Risk–Based Modeling, Simulation and Optimization for the Integration of Renewable Distributed Generation Into Electric Power Networks

THÈSE présentée par

Rodrigo MENA

pour l’obtention du

GRADE DE DOCTEUR

Spécialité: Génie Industriel

Laboratoire d’accueil: Laboratoire de Génie Industriel

SUJET:

Risk–based Modeling, Simulation and Optimization for the Integration of Renewable Distributed Generation into Electric Power Networks

Modélisation, simulation et optimisation basée sur le risque pour l’intégration de génération distribuée renouvelable dans des réseaux de puissance électrique

soutenue le: 30 juin 2015

devant un jury composé de:

Enrique DROGUETT University of Maryland Reviewer Federica FOIADELLI Politecnico di Milano Reviewer Michel MINOUX Université Pierre et Marie CURIE Examiner Guillaume SANDOU CentraleSupélec Examiner Vera SILVA Électricité de France R&D Examiner Enrico ZIO CentraleSupélec Supervisor ❖ Martin HENNEBEL CentraleSupélec Co–supervisor Yan–Fu LI CentraleSupélec Co–supervisor

2015ECAP0034

To my brother . . .

Acknowledgements

I would like to acknowledge my thesis supervisor, Professor Enrico Zio, for having given me the opportunity to develop this Ph.D. work. Thanks for trusting me, for all the guidance and advice to improve my knowledge of how to do forefront research, for constantly spreading enthusiasm and eagerness in learning and for having encouraged me to overcome the difficulties encountered over these three years. To me, It has been a privilege to work in Prof. Zio’s team and undoubtedly a real fruitful experience, both professionally and personally.

I extend my appreciation to my co–supervisors, Dr. Yan–fu Li and Dr. Martin Hennebel, and to a former collaborator of the Chair SSEC, Dr. Carlos Ruiz, whose generous support has been fundamental in the development of this work. I am grateful to you for having been always interested and attentive in following the progress of this thesis, for the willingness and patience to discuss and teach me whenever I needed technical counsel.

To all the members of the jury, Professors Federica Foiadelli, Guillaume Sandou, Michel Minoux and Enrique Droguett, and Dr. Vera Silva, my sincere gratitude for your time and consideration. It was an absolute challenge having defended my thesis in front of such excelled academics and professionals. I thank you all for the constructive remarks, comments and suggestions and, in particular, Prof. Foiadelli and Prof. Droguett that kindly accepted to be the reviewers of the manuscript.

My warmest gratitude and recognition go to Corinne Ollivier, Delphine Martin and Sylvie Guillemain, the three loving assistants (a.k.a. supreme bosses) of the Industrial Engineering Department (LGI), because of their unconditional good will to help us, the usually lost–in–translation Ph.D. students. In addition, I would like to thank my friends and colleagues from LGI, for all the unforgettable moments.

To conclude, words are not enough to express how beholden I am to my beloved parents, siblings and Elisa, they just know that I am.

Abstract

Renewable distributed generation (DG) is expected to continue playing a fundamental role in the development and operation of sustainable, efficient and reliable electric power systems, by virtue of offering a practical alternative to diversify and decentralize the overall power generation, benefiting from cleaner and safer energy sources. The integration of renewable DG in the existing electric power networks poses socio–techno–economical challenges, which have attracted substantial research and advancement.

In this context, the focus of the present thesis is the design and development of a modeling, simulation and optimization framework for the integration of renewable DG into electric power networks. The specific problem considered is that of selecting the technology, size and locationof renewable generation units, under technical, operational and economic constraints. Within this problem, key research questions to be addressed are: (i) the representation and treatment of the uncertain physical variables (like the availability of diverse primary renewable energy sources, bulk– power supply, power demands and occurrence of components failures) that dynamically determine the DG–integrated network operation, (ii) the propagation of these uncertainties onto the system operational response and the control of the associated risk and (iii) the intensive computational efforts resulting from the complex combinatorial optimization problem of renewable DG integration.

For the evaluation of the system with a given plan of renewable DG, a non–sequential Monte Carlo simulation and optimal power flow (MCS–OPF) computational model has been designed and implemented, that emulates the DG–integrated network operation. Random realizations of operational scenarios are generated by sampling from the different uncertain variables distributions, and for each scenario the system performance is evaluated in terms of economics and reliability of power supply, represented by the global cost (CG) and the energy not supplied (ENS), respectively. To measure and control the risk relative to system performance, two indicators are introduced, the conditional value–at–risk (CVaR) and the CVaR deviation (DCVaR).

For the optimal technology selection, size and location of the renewable DG units, two distinct multi–objective optimization (MOO) approaches have been implemented by heuristic optimization (HO) search engines. The first approach is based on the fast non–dominated sorting genetic algorithm (NSGA–II) and aims at the concurrent minimization of the expected values of CG and ENS, then ECG and EENS, respectively, combined with their corresponding CVaR(CG) and CVaR(ENS) viii values; the second approach carries out a MOO differential evolution (DE) search to minimize simultaneously ECG and its associated deviation DCVaR(CG). Both optimization approaches embed the MCS–OPF computational model to evaluate the performance of each DG–integrated network proposed by the HO search engine.

The challenge coming from the large computational efforts required by the proposed simulation and optimization frameworks has been addressed introducing an original technique, which nests hierarchical clustering analysis (HCA) within a DE search engine.

Examples of application of the proposed frameworks have been worked out, regarding an adaptation of the IEEE 13 bus distribution test feeder and a realistic setting of the IEEE 30 bus sub–transmission and distribution test system. The results show that these frameworks are effective in finding optimal DG–integrated networks solutions, while controlling risk from two distinct perspectives: directly through the use of CVaR and indirectly by targeting uncertainty in the form of DCVaR. Moreover, CVaR acts as an enabler of trade–offs between optimal expected performance and risk, and DCVaR integrates also uncertainty into the analysis, providing a wider spectrum of information for well–supported and confident decision making.

The main original contributions of the thesis work here presented reside in: framing the problem of optimal technology selection, size and location of renewable generation units, within an integrated simulation and optimization approach that takes into consideration multiple uncertain operational inputs through the developed MCS–OPF, allows assessing and controlling risk by introducing CVaR and DCVaR measures, and copes with computational complexity by embedding HCA into the HO search engine.

Keywords: renewable distributed generation, uncertainty, risk, simulation, optimization, conditional value–at–risk, conditional value–at–risk deviation, genetic algorithm, differential evolution, hierarchical clustering analysis Résumé

Il est prévu que la génération distribuée par l’entremise d’énergie de sources renouvelables (DG) continuera à jouer un rôle clé dans le développement et l’exploitation des systèmes de puissance électrique durables, efficaces et fiables, en vertu de cette fournit une alternative pratique dedécen- tralisation et diversification de la demande globale d’énergie, bénéficiant de sources d’énergie plus propres et plus sûrs. L’intégration de DG renouvelable dans les réseaux électriques existants pose des défis socio–technico–économiques, qu’ont attirés de la recherche et de progrès substantiels.

Dans ce contexte, la présente thèse a pour objet la conception et le développement d’un cadre de modélisation, simulation et optimisation pour l’intégration de DG renouvelable dans des réseaux de puissance électrique existants. Le problème spécifique à considérer est celui de la sélection de latech- nologie, la taille et l’emplacement de des unités de génération renouvelable d’énergie, sous des con- traintes techniques, opérationnelles et économiques. Dans ce problème, les questions de recherche clés à aborder sont: (i) la représentation et le traitement des variables physiques incertains (comme la disponibilité de les diverses sources primaires d’énergie renouvelables, l’approvisionnements d’électricité en vrac, la demande de puissance et l’apparition de défaillances de composants) qui déterminent dynamiquement l’exploitation du réseau DG–intégré, (ii) la propagation de ces incerti- tudes sur la réponse opérationnelle du système et le suivi du risque associé et (iii) les efforts de calcul intensif résultant du problème complexe d’optimisation combinatoire associé à l’intégration de DG renouvelable.

Pour l’évaluation du système avec un plan d’intégration de DG renouvelable donné, un modèle de calcul de simulation Monte Carlo non–séquentielle et des flux de puissance optimale (MCS–OPF) a été conçu et mis en œuvre, et qui émule l’exploitation du réseau DG–intégré. Réalisations aléatoires de scénarios opérationnels sont générés par échantillonnage à partir des différentes distributions des variables incertaines, et pour chaque scénario, la performance du système est évaluée en termes économiques et de la fiabilité de l’approvisionnement en électricité, représenté par le coût global (CG) et l’énergie non fournie (ENS), respectivement. Pour mesurer et contrôler le risque par rapport à la performance du système, deux indicateurs sont introduits, la valeur–à–risque conditionnelle (CVaR) et l’écart du CVaR (DCVaR).

Pour la sélection optimale de la technologie, la taille et l’emplacement des unités DG renouvelables, deux approches distinctes d’optimisation multi–objectif (MOO) ont été mis en œuvre x par moteurs de recherche d’heuristique d’optimisation (HO). La première approche est basée sur l’algorithme génétique élitiste de tri non-dominé (NSGA–II) et vise à la réduction concomitante de l’espérance mathématique de CG et de ENS, dénotés ECG et EENS, respectivement, combiné avec leur valeurs correspondent de CVaR(CG) et CVaR(ENS); la seconde approche effectue un recherche à évolution différentielle MOO (DE) pour minimiser simultanément ECG et s’écart associé DCVaR(CG). Les deux approches d’optimisation intègrent la modèle de calcul MCS–OPF pour évaluer la performance de chaque réseau DG–intégré proposé par le moteur de recherche HO.

Le défi provenant de les grands efforts de calcul requises par les cadres de simulation et d’optimisation proposée a été abordée par l’introduction d’une technique originale, qui niche l’analyse de classification hiérarchique (HCA) dans un moteur de recherche deDE.

Exemples d’application des cadres proposés ont été élaborés, concernant une adaptation du réseau test de distribution électrique IEEE 13–nœuds et un cadre réaliste du système test de sous– transmission et de distribution IEEE 30–nœuds. Les résultats montrent que les cadres proposés sont efficaces dans la recherche des solutions de réseaux DG–intégrés optimales, tout en contrôlant les risques à partir de deux perspectives distinctes: directement par l’utilisation du CVaR et indirectement par en ciblant l’incertitude sous la forme du DCVaR.

Les principales contributions originales de la thèse présentée ici résident dans: encadrer le problème de la sélection optimale de la technologie, la taille et l’emplacement des unités de génération renouvelable d’énergie, dans une approche intégrée de simulation et d’optimisation qui tient compte des multiples variables opérationnelles incertains à travers du MCS–OPF développé, permet d’évaluer et contrôler le risque en introduisant les mesures CVaR et DCVaR, et fait face à la complexité de calcul en intégrant HCA dans le moteur de recherche HO.

Mots clés: génération distribuée, énergie renouvelable, incertitude, risque, simulation, optimisation, valeur–à–risque conditionnelle, écart du valeur–à–risque conditionnelle, algorithme génétique, évolution différentielle, analyse de classification hiérarchique Table of contents

List of figures xv

List of tables xvii

Nomenclature xix

I Thesis 1

1 Introduction 3

1.1 Electric power systems ...... 3

1.1.1 Transmission & distribution of electric power ...... 5

1.1.2 Diversification and decentralization of electrical power generation .6

1.2 Renewable distributed generation ...... 7

1.2.1 Potential benefits and challenges ...... 7

1.3 Research objectives and original contributions ...... 10

1.4 Thesis structure ...... 12

2 Renewable DG–integrated electric power network modeling 15

2.1 System representation ...... 15

2.1.1 Components classification ...... 15

2.1.2 Network topology ...... 16

2.2 Uncertain operational inputs ...... 18

2.2.1 Power demands and energy price ...... 18 xii Table of contents

2.2.2 Bulk–power supply ...... 20

2.2.3 Solar photovoltaic generation ...... 20

2.2.4 Wind turbines generation ...... 21

2.2.5 Electrical vehicles ...... 22

2.2.6 Storage devices ...... 23

2.2.7 Components availability state ...... 24

2.3 Stochastic operation modeling ...... 24

2.3.1 Non–sequential Monte Carlo simulation ...... 24

2.3.2 Optimal power flow ...... 25

2.4 Performance evaluation ...... 27

2.4.1 Global cost ...... 27

2.4.2 Energy not supplied ...... 28

2.4.3 Uncertainty and risk measurement ...... 28

3 Renewable DG–integrated electric power network planning 31

3.1 Optimal DG technologies selection, sizing and allocation ...... 31

3.1.1 Optimization strategies formulation ...... 31

3.2 Heuristic optimization & MCS–OPF simulation frameworks ...... 34

3.2.1 Non–dominated sorting genetic algorithm II–based approach ...... 35

3.2.2 MOO differential evolution–based approach ...... 39

4 Computational Challenge 43

4.1 Clustering in heuristic optimization ...... 43

4.2 Hierarchical clustering & differential evolution ...... 44

5 Applications 51

5.1 A risk–based MOO and MCS–OPF simulation framework for the integration of renewable DG and storage devices ...... 51

5.1.1 IEEE 13–bus test feeder case ...... 51

5.1.2 Conditional value–at–risk: expected performance and risk trade–off ...... 52 Table of contents xiii

5.1.3 Optimal DG–integrated network plans ...... 55

5.1.4 Brief summary ...... 56

5.2 Hierarchical clustering analysis and differential evolution (HCDE) for optimal integration of renewable DG ...... 57

5.2.1 IEEE 13–bus test feeder case ...... 57

5.2.2 Quantification of the benefits of HCDE ...... 58

5.2.3 Identification of time complexity conditions and limits ...... 61

5.2.4 Optimal DG–integrated network plans ...... 63

5.2.5 Brief summary ...... 64

5.3 A MOO and MCS–OPF simulation framework for risk–controlled integration of DG 65

5.3.1 IEEE 30 bus sub-transmission and distribution test system ...... 65

5.3.2 Conditional value–at–risk deviation: expected performance, uncertainty and risk trade–off ...... 67

5.3.3 Optimal DG–integrated network plans ...... 71

5.3.4 Brief summary ...... 73

6 Conclusions 75

6.1 Original contributions ...... 75

6.2 Future work ...... 77

Appendix A IEEE 13–bus test feeder data: Application 5.1 79

Appendix B IEEE 13–bus test feeder data: Application 5.2 83

Appendix C IEEE 30–bus sub–transmission & distribution system data: Application 5.3 87

References 91

II Papers 99

List of figures

1.1 World CO2 emissions by sector [1] ...... 4 1.2 Example of a passive T&D network ...... 5

1.3 Example of an active T&D network ...... 6

1.4 Thesis structure diagram ...... 13

2.1 IEEE 30 bus sub–transmission and distribution test system diagram ...... 16

2.2 Example of a nodal daily load profile, hourly normally distributed ...... 19

2.3 Proportional correlation energy price vs aggregated load ...... 19

2.4 Example of hourly probability distribution of EV operating states per day ...... 22

2.5 Sampling process ...... 25

2.6 Graphic representation of VaRα(x), CVaRα(x) and DCVaRα(x); x = loss ...... 28

3.1 Flow chart of the proposed NSGA–II MCS–OPF framework ...... 38

3.2 Flow chart of the proposed MOO–DE MCS–OPF framework ...... 41

4.1 Example dendrogram for average linkage HCA ...... 46

4.2 Example of cutoff distance calculation ...... 47

4.3 Flow chart of the proposed HCDE framework ...... 49

5.1 Radial 11–nodes distribution network ...... 52

5.2 Pareto fronts for different values of β ...... 53

5.3 Bubble plots EENS v/s ECG. Diameter of bubbles proportional to CVaR(ENS) (A) and CVaR(CG) (B)...... 53 xvi List of figures

5.4 Average total DG power allocated (A) and its breakdown by type of DG: PV (B), W (C), EV (D) and ST (E) ...... 55

5.5 Radial 11-nodes distribution network ...... 58

5.6 ECGmin vs NFE for NP 10, 20, 30, 40, 50 set in DE ...... 58 ∈ { } 5.7 ECGmin vs NFE for each (NP, CCCT , pco) set in HCDE ...... 59 5.8 CCC behavior per generation g ...... 60

g 5.9 Empirical NP pdf for each (NP, CCCT , pco) set in HCDE ...... 60

5.10 Average total DG power allocated and investment cost for representative (NP, CCCT ,

pco) settings ...... 63

5.11 Nodal average total DG power for representative (NP, CCCT , pco) settings ...... 64 5.12 IEEE 30 bus sub-transmission and distribution test system diagram ...... 66

5.13 Set of non-dominated solutions: Pareto front ...... 67

5.14 Set of non–dominated solutions with iso–CVaR (A) and iso–DCVaR (B) curves . . . . 68

5.15 Empirical CG probability density functions ...... 69

5.16 Total average renewable power installed. Cases ECGmin (A), CVaRCGmin (B) and DCVaRCGmin (C)...... 71

5.17 Nodal average power by type of generator ...... 72

A.1 Mean (A) and variance (B) values of nodal power demand daily profiles ...... 79

A.2 Hourly per day probability data of EV operating states. ρ = 1: charging, ρ = 0: − disconnected, ρ = 1: discharging ...... 81

B.1 Mean and standard deviation values of normally distributed nodal power demand daily profile ...... 83

B.2 Hourly per day probability data of EV operating states ...... 84

C.1 Accumulated mean (A) and standard deviation (B) values of nodal load daily profiles 87 List of tables

4.1 Example hierarchical structure outcome ...... 45

5.1 Objective functions: expected and CVaR values of selected Pareto front solutions . . . 54

5.2 Average, minimum and maximum total DG power allocated per node ...... 56

5.3 Asymptotic time complexity of the algorithms ...... 62

5.4 Ratio κ for each (NP, CCCT , pco)...... 63 5.5 MOO-DE and MCS–OPF parameters ...... 65

5.6 Ratio of power usage by type of generator ...... 70

A.1 Feeders characteristic and technical data [2] ...... 79 A.2 Bulk–power supply parameters ...... 80

A.3 Parameters of PV,W, EV and ST technologies [3–5] ...... 80

A.4 Failure rates of feeders, MG and DG units [3–6] ...... 80

A.5 Investment, fixed O&M and variable O&M costs of MG andDG [6–8] ...... 81

B.1 Feeders characteristic and technical data [2,4,9 ] ...... 83

B.2 Power sources parameters and technical data [3–8, 10] ...... 84

C.1 T&D lines characteristics and technical data [11–13] ...... 88

C.2 Power generators technical data and uncertain model parameters [3–5,7, 10, 14 ] .. 89

C.3 Power generators failure and repair rates and costs [3,4,7, 10, 14, 15 ] ...... 89

Nomenclature

Roman Symbols BGT Budget ($)

B(i,i') Susceptance of T&D line (i, i') (p.u.) CCC Cophenetic correlation coefficient

CCCT Cophenetic correlation coefficient threshold cdf Cumulative distribution function CG Global cost ($/h) G Ω G ω CG Sample of NS realizations of CGs ($) GTD Ω GTD ω CG Sample of NS realizations of CGs ($/h) CI j + CO f j Fixed investment and operating cost ($) CLS Load shedding cost ($/MWh) Group (cluster) of individuals GC Ω G ω CO Sample of NS realizations of COs ($) GTD Ω GTD ω CO Sample of NS realizations of COs ($/h)

COv(i,i') Variable operating cost of the T&D line (i, i') ($/MWh) COvj Variable operating cost of the power generator type j ($/MWh) D Set of hours of the day ¯1 1 D Average of the Euclidean distances dı, dC Crowding distance

DL Daylight interval z dmin Minimum linkage distance dı, dNC=4 Distance to form at least four clusters ∆t Time interval (h) Dz Matrix of linkage distances z D Linkage distance between groups ı and  (a) ECG Expected global cost (FW (a))C C ($) (b,c) ECG Expected global cost (FWs (b) & (c)) ($/h) EENS Expected energy not supplied (MW h) ℓ Non–dominated front index xx Nomenclature

ENS Energy not supplied (MW h) Ω ω ENS Sample of NS realizations of ENSs (MW h) EPmax Maximum value of energy price ($/MWh) EPΩ Sample of NS realizations of EP , ω Ω ($ MWh) ts s / ∀ ∈ EPt Energy price at hour of the day t ($/MWh) EV Electrical vehicles type of DG F Differential variation amplification factor

f j,t (ρj,t ) Hourly pdf of operational states of EV type j FFj Fill factor fi,t (Li,t µi,t , σi,t ) Normal pd f of nodal power load at hour of the day t | f j(Pj µj, σj) Normal pdf of bulk–power supply MG type j | fi(Hi αi, βi, pL, Hi∗) Adjusted solar irradiance Beta pdf f J ,|J , M Uniform pdf of ST type j j( Sj Tj ) | fi(Ui α∗i , β∗i ) Wind speed Weibull pdf G | Class of power generation components G ω COs Aggregated operating cost for scenario ωs ($) GTD ω COs Aggregated operating cost for scenario ωs ($/h) Hi Solar irradiance at node i

Hi∗ pL%ile of the non–adjusted solar irradiance Beta pdf at node i I Current at maximum power point of PV type j (A) MPPj inc Incentive for power generation from DG sources ($/MWh) I Short circuit current of PV type j (A) SCj J Level of charge in the storage device (MJ) J Specific energy of the active chemical in ST type j (MJ kg) Sj / fk Availability states stationary pdf of component k k Current temperature coefficient of PV type j (mA C) I j /◦ L Power demand (MW)

z∗ lk,k' Linkage distance dı, where Xk and Xk' become members of ı, C ¯ Average of the linkage distances lk,k'

L z∗ z∗ Matrix of linkage distances lk,k' = dı, L Li Active power load at node i (MW) L Maximum value of power load at node i (MW) maxi LSi,s Load shedding at node i for scenario ωs (MW) Ω LSi Sample of NS realizations of LSi,s (MW h) m Total number of types of bulk–power suppliers MG Subclass of bulk–power generation components M Mass of active chemical in ST type j (kg) Tj N Set of all nodes in the electric power network nc Number of photovoltaic cells Nomenclature xxi

Non–dominated front FNF Number of objective functions NFE Number of objective function evaluation

NGmax Maximum number of generations NP Population size NS Number of operational scenarios realizations NZ Number of steps p−j,t Hourly pdf of charging operational state of EV type j pC Crossover probability pco Cutoff level coefficient + pj,t Hourly pdf of discharging operational state of EV type j pdf Probability density function PF DG DG penetration factor PF DG Maximum allowed penetration factor of DG technology type j max j P Active power injected or generated at node i (MW) Gi Pi Active power leaving node i (MW)

Pi,j Power output of generator type j at node i (MW)

Pj,t Power output of generator type j at hour of the day t (MW)

Pj Available bulk–power supply MG type j (MW) pL Probability that the hour of the day t is in the interval DL pM Mutation probability P Power rating of the T&D line i, i' (MW) max(i,i') ( ) P Maximum power capacity of generator type j (MW) max j PΩ Sample of NS realizations of P , ω Ω (MW) Ui,j Ui,j,s s POP Population ∀ ∈ P Rated power of generator type j (MW) R j P Used power from the generator type j (MW) Ui,j,s PV Solar photovoltaic type of DG 0 pj,t Hourly pdf of disconnected operational state of EV type j Q Matrix of location and capacity size of power generation units qi,j Number of power generation units type j allocated at node i QM Matrix of location and capacity size of MG q' Non–negative integer QR Matrix of location and capacity size of renewable DG r Total number of types of renewable technologies RG Subclass of renewable DG components

Sre f Reference apparent power in the network (MV A) ST Storage devices type of DG t Hour of the day xxii Nomenclature

T Ambient temperature at node i ( C) Ai ◦ t∆ Operational scenarios of duration i (h) TD Class of transmission and distribution lines components t H Lifetime of the project (h) J t j Upper bound of discharging time interval of ST type j (h) T Nominal cell operation temperature of PV type j ( C) N oj ◦ ρ t Time of residence in the operating state ρj,t (h) U Average wind speed of W type j (m s) Aj / U Cut–in wind speed of W type j (m s) CI j / U Cut–out wind speed of W type j (m s) COj / Ui Wind speed at node i (m/s) k Voltage temperature coefficient of PV type j (mV C) Vj /◦ V Voltage at maximum power point of PV type j (V) MPPj V Open circuit voltage of PV type j (V) OCj W Wind turbines type of DG Y Set of transmission and/or distribution lines

Z∗ Set of non–negative integers Greek Symbols

αi, βi Shape parameters of Beta distribution at node i

α∗i , β∗i Scale and Shape parameters of Weibull distribution at node i α Confidence level Γ Gamma function

ηk Availability state of component k λ Failure rate of component k (n h) Fk / λ Repair rate of component k (n h) Rk / µi,t Mean of nodal power load at hour of the day t (MW)

µj Mean of bulk–power supply MG type j (MW) Ω Set of all the NS realizations of operational scenarios ω ω Set of sampled variables or operational scenarios Φ Standard Normal cdf φ Standard Normal pdf

ρj,t Possible operational states of EV type j

σi,t Standard deviation of nodal power load at hour of the day t (MW)

σj Standard deviation of bulk–power supply MG type j (MW)

τj Maximum number of units of DG technology j to allocate

δi Voltage angle at node i Subscripts g Generation index i Node index Nomenclature xxiii

ı Group (cluster) of individuals index j Power generator type index  Group (cluster) of individuals index k Component index k Individual index k' Individual index s Scenario index t Hour of the day index z Steps index Acronyms / Abbreviations CHP Combined heat and power CLLI Contingency load loss index

CO2 Carbon dioxide CVaR Conditional value–at–risk DCVaR Conditional value–at–risk deviation DE Differential evolution DG Distributed generation ECOST Expected value of non–distributed energy cost EENS Expected energy not supplied FW Framework HCA Hierarchical clustering analysis HCDE Hierarchical clustering differential evolution HO Heuristic optimization LHS Latin Hypercube Sampling MCS Monte Carlo simulation MINLP Mixed–integer non–linear problem MOO Multi–objective optimization NSGA–II Fast non–dominated sorting genetic algorithm O&M operation and maintenance OPF Optimal power flow SAIDI System average interruption duration index SAIFI System average interruption frequency index SO Single objective T&D Transmission and distribution TPES Total primary energy supply TVD Total voltage deviation VaR Value–at–risk

I Thesis

1 Introduction

The present thesis describes the works done in the design and development of a modeling, simulation and optimization framework for the integration of renewable distributed generation (DG) into electric power networks. Specifically, the problem considered is that of technology selection, sizing and allocation of renewable DG units, taking into account uncertainty and risk.

This introductory chapter is organized as follows. Section 1.1 acquaints the reader with the research context, presenting the motivations for advancements in sustainable electric power systems and the role that renewable DG plays in the diversification and decentralization of power generation. Section 1.2 discusses the potential benefits of DG and the major challenges involved in its integration into existing power networks. In Section 1.3, key research questions and ensuing objectives are formulated in the effort of contributing to shape frameworks for well–supported decision–making in optimal DG integration. Finally, a schematic representation of the overall structure of the thesis and general descriptions of the developed frameworks are provided in Section 1.4.

1.1 Electric power systems

The adverse environmental effects accompanying the intensive and prolonged use of fossil fuels are not anymore a midterm conjecture but an ongoing reality. Even though initiatives with considerable participation, like the Kyoto Protocol and the Copenhagen Accord, have set targets to limit carbon dioxide (CO2) emissions in order to constrain the global temperature rise to less than 2◦C relative to the pre–industrial level between 2008–12, no overall mitigation has been achieved [1]. Indeed, greenhouse gas emissions from fuel combustion have been steadily increasing during the last decade, mainly by cause of the accelerated economic and population growths of industrialized countries, that do not face emission targets and, consequently, demand progressively larger amounts of energy from fossil fuels.

Given the above, the development of environmental sustainability has become a worldwide imperative for all sectors of highly compromised human activities such as industry, transport, residential and electricity and heat generation. In particular, the electric power generation sector is 4 1 Introduction

considered an important point of concern, since it subscribes 42% of the global CO2 emissions, by far being the largest contribution, as shown in Figure 1.1 [1].

Other 9% Electricity Transport & 23% heat 42%

Residential 6% Industry 20%

Figure 1.1 World CO2 emissions by sector [1]

In the pursuit of sustainable energy systems, the heightening of regulatory targets and the engagement of leading–role countries to participate in international agreements certainly set a challenging framework for advancement, but these measures do not constitute solutions by them- selves. Nevertheless, the awareness on the global environmental problem has triggered a revolution towards cleaner, safer and more reliable systems along all the energy value chain [10, 15–17]. Electric power systems are the ‘spinal cord’ of the energy value chain and they are facing a stimulating transition across all their three main components, generation, transmission and distribution, led by both technological development of new equipment and devices and enhanced actions in planning, operation and management strategies and driven by the opportunity of generating electrical power by making use of low–carbon and, in preference, renewable energy sources. This offers a great opportunity to overcome the growing energy demand, mitigating greenhouse emissions and alleviating the energy market instability associated to the depletion of fossil fuels [18]. The integration of renewable generation into power systems is implemented mainly in two ways, depending on the scale of the available primary energy sources: large–scale renewable generation is predominantly connected upstream sub–transmission and distribution networks, acting as a conventional bulk–power plant, while small–scale renewable generation units are allocated close or directly on the ‘customers site of the meter’, which is known as distributed generation (DG) [19–21]. Both plan of actions offer the advantage of generating clean electric power, but it is particularly DG that is playing a crucial role because it provides the possibility of concurrently diversifying and decentralizing the overall power generation.

The connection of diverse renewable DG units onto sub–transmission and distribution networks implies conceptual and operational transformations which are explained in the next section. 1 Introduction 5

1.1.1 Transmission & distribution of electric power

Transmission and distribution (T&D) of electric power has been traditionally a passive ‘fit–and– forget’ strategy, characterized by unidirectional power flows supplied by centralized generation systems [16, 19, 22, 23]. Moreover, the restrictive structure of conventional T&D settings makes it difficult to supply power to remote areas, due to the extra T&D expenses associated andthe risk of compromising a large portion of power supply due to the occurrence of outages upstream sub–transmission and distribution networks [24, 25].

Generation conventional power plant

Residential customers

Transmission

Distribution

Industrial customers

power supplied by conventional generators C switch

Figure 1.2 Example of a passive T&D network

Figure 1.2 illustrates the constricted configuration of a typical passive T&D network. Itcan be noticed that any contingency upstream point A may produce a large loss of power supply, in this example, affecting mainly the industrial customers given the radial topology of that portion of the network. For residential customers, the impact of an outage level at points A or B can be deaden by closing the switch at point C, which provides a meshed character to that portion of the network and, therefore, some level of redundancy. Notwithstanding mesh–structured T&D networks are more flexible in terms of reliability of power supply, their operation is more expensive and complicated since infrastructure and operation and maintenance costs (O&M) of T&D lines increase, more control and protection devices are needed with the respective synchronization in their operation [9, 26, 27]. 6 1 Introduction

1.1.2 Diversification and decentralization of electrical power generation

On a large scale, hydro–power technology is the major contributor to carbon–free generation, representing the 2.4% of the global total primary energy supply (TPES) in 2012 [28]. Over decades, it has played an important environmental role but current expansions of hydro–power capacity are negligible with respect to other pollutant technologies like oil, coal, gas, bio–fuels and waste, which state the 31.4, 29.0, 21.3, 10.0% of the TPES in 2012 [28], respectively, and account for the 99% of the global CO2 emissions [1]. Indeed, global hydro–power additions decreased in 2013 [29], indicating a loss of priority in mitigating CO2 emissions by use of this technology. Moreover, large scale hydro–power plants operate under the conventional passive logic, then, no further decentralization can be achieved from them.

Generation conventional power plant Residential customers

Transmission

Industrial customers A

Distribution

solar photovoltaic generation power supplied by conventional generators wind turbines power supplied by renewable DG electric vehicle C switch storage device

Figure 1.3 Example of an active T&D network

It is the rapid expansion of non–hydro renewable capacity, that in 2014 rose globally by ap- prox. 7% (350 (TW h))[29], that is making a difference. With the integration of renewable DG technologies, such as small–scale hydro, solar photovoltaic, on and off–shore wind and storage devices, electric power systems are evolving towards an active operational strategy, with possibly bidirectional power flows and, thus, more reliable, decentralized and diversified sources [16, 19], 1 Introduction 7 as shown in Figure 1.3. In the Figure, the impact on the power supply of outages at points A and/or B is smoothed by the deployment of different renewable DG units throughout the T&D network.

Finally, renewable DG technologies make use of local/regional renewable energy sources, which boosts them in view of the requested environmental sustainability, while engaging the participation of households, small businesses and specialized power companies and offering interesting techno– economical benefits.

1.2 Renewable distributed generation

In the literature, DG is commonly defined as modular (independent) generation units located ator in the neighborhood of power demand spots, connected to sub–transmission and/or distribution networks rather than high voltage transmission [19–21, 27, 30]. The modular characteristic refers to their smaller scale power capacity, generally in the range of generation 1 (kW)–5 (MW) [25]. It is important to mention that some DG applications can reach a considerable power capacity, between 50 and 300 (MW); however, there is disagreement among authors about at what extent these can be considered as DG.

The technological spectrum of DG is formed by two main groups: carbon–free and low–carbon ‘efficient’ generation devices. The first group includes small hydro and tidal turbines, solarpho- tovoltaic panels, wind turbines, geothermal steam–turbines, etc., whereas the second one adds biomass, waste and combined heat and power (CHP) devices and installations, based on recipro- cating engines and combustion gas turbines [19]. Moreover, storage devices and electric vehicles are also feasible technologies accompanying DG integration, to increase the overall efficiency of activated electric power networks [6, 19, 31, 32]. Without loss of generality, in the present thesis work the focus of attention is oriented to carbon– free DG given their superior contribution to environmental sustainability. Below, the potential technical and economical benefits on one side and the possible operational complications onthe other are discussed.

1.2.1 Potential benefits and challenges

Under the assumption that DG power is ‘dispatchable’, i.e., DG units are able to provide and sell energy in parallel to or as competitors of the centralized bulk–power supply to satisfy the system demand, and given the fact that by integrating DG into an existing network the power flows through shorter and possibly bidirectional paths, the main technical benefits that can be achieved are [3,4,9, 10, 16, 17, 19, 22, 33–44 ]:

Improvement of reliability of power supply. ◦ 8 1 Introduction

Reduction of power losses. ◦ Voltage stability. ◦ Enhancement of power quality. ◦ Alleviation of T&D lines congestion. ◦ Power supply autonomy of rural/isolated areas. ◦ The most representative economical benefits expected from the integration of renewable DGare the following [4, 10, 16, 17, 19, 23, 34, 35, 38–41, 43, 45, 46]:

Deferral of investments for conventional power plants and T&D upgrades. ◦ Reduction of T&D O&M costs. ◦ Decrement of generation costs and energy price. ◦ Reduction of fossil fuel costs. ◦ Diminution of price volatility associated to fossil fuels. ◦ Reduction of investment risks. ◦ However, planners and operators have to face complex economic, regulatory, technical and operational constraints, within which the dynamics introduced by renewable DG may generate complications that can counteract the potential benefits [25, 26, 35]. Indeed, the traditional passive power systems structures are designed to operate with unidirectional power flows, and in order to incorporate renewable DG, the need for more protection and control devices may be significant. Reactive power may also result compromised since many renewable DG technologies supply only active power, affecting also the voltage stability and the introduction of harmonics [19]. The above–mentioned challenges have prompted substantial research and development. In particular, DG planning has been a fundamental baseline of advancement to properly seize its potential advantages. The specific problem associated to DG planning consists in selecting the technology, size and location of renewable generation units while respecting the techno–economical, regional and regulatory constraints imposed by the existing system.

One of the main difficulties associated to DG planning is the proper modeling of the intrinsic uncertain behavior of primary renewable energy sources (e.g. solar irradiance, wind speed and water inflow, in the case of solar photovoltaic, wind turbines and hydro–power technologies, respectively) and of the stochastic occurrence of unexpected events on the DG units, such as failures and stoppages, that may interrupt or curtail the power generation capacity. Indeed, these sources of uncertainty come on top of those already present in the operating power systems, outage events due to failures (or stoppages) of T&D lines and/or conventional power generators, variability and growth of power demand, volatility in the energy price and fluctuations in the bulk–power supply, among others. As 1 Introduction 9 a consequence, for any proposed DG–integrated network plan, the uncertain operational conditions need to be considered in the system response as calculated by solving power flow equations under a number of representative scenarios [19, 42, 43, 47]. The decision making for DG–integrated network planning is usually framed as an optimization problem, which can be especially complex depending on the size and topology of the network, the number of load nodes, number of available DG technologies and the aforementioned uncertainty and the non–linear conditions arising from technical constraints [16, 20, 42]. Optimality of the renewable DG plan is customarily sought with regards to cost–based, operational or technical targets. Among cost–based objectives are the costs of energy and fuel for generation, investments,

O&M, energy purchased from conventional power plants, energy losses, CO2 emissions, taxes, incentives, incomes, etc. [3, 4, 7, 9, 10, 15, 16, 19, 22, 38, 48–57]. Operational objective functions consider the performance and reliability of power supply in terms of indicators like the expected energy not supplied (EENS) [37, 58], contingency load loss index (CLLI) [9], expected value of non–distributed energy cost (ECOST) , system average interruption frequency index (SAIFI), system average interruption duration index (SAIDI) [16, 22, 50], among others. Concerning technical targets, the most commonly used indexes are the total voltage deviation (TVD) [51] and energy losses [10, 44]. Obviously, the optimal solutions must also comply with the system technical constraints like generation capacities, T&D lines rating and voltage drops [19, 25]. In the search for optimal DG–integrated network configurations under uncertainty, the use of only expected or cumulative indicators as objective function(s) hinders the possibility of controlling the risk associated to the optimal solution(s): the expected performance of an optimal DG–integrated network may be satisfactory but be exposed to high variability or to risky scenarios with non– negligible probabilities.

One way to account for uncertainty and risk is to frame DG planning as a portfolio optimization problem, in which the different types of DG technologies are treated analogously to financial assets [42, 46, 59–65].

In portfolio optimization theory, the mean–variance approach is the most common [66] and has been applied to DG planning also [42, 46, 59, 62, 65, 67]. However, from a risk–perspective it entails a drawback that cannot be ignored: the variance measure includes the values of performance that symmetrically fall short of or exceed the expected or mean value; in the search for optimal DG technologies portfolios, lower levels of uncertainty (variance) in the performance function can be obtained with portfolios that lead to rarer occurrences of both beneficial and/or non–desired (risky) scenarios. Then, for controlling the risk side, it is necessary to introduce additional indicators that provide information on the extent of asymmetry of the performance function, weighting accordingly the risky part of it, e.g., by skewness and kurtosis indicators that estimate the asymmetry and peakedness of a probabilistic performance function, respectively [68]. 10 1 Introduction

Likewise derived from portfolio optimization theory, direct risk–based frameworks have been formulated and applied to address DG planning problems under uncertain conditions. The most widely used risk measures are the value–at–risk (VaR) and conditional value–at–risk (CVaR) [61, 63, 64, 69–73], which focus on non–desirable performance outcomes given by a portfolio relative to a specific confidence level or percentile.

Even though integrated portfolio optimization approaches, for instance mean–variance–skewness or mean–variance–CVaR, present robustness advantages by conjointly controlling the level of uncertainty and risk associated to the expected value of performance of different portfolios, they can considerably increase the complexity of the concurrent optimization problem. Considering the expected or mean value and the necessary deviation and risk measures as representative objectives of a portfolio increases significantly the number of objective functions to be simultaneously optimized, further constraining the feasible space and, eventually, hindering the understanding of the information delivered to the decision–makers.

Moreover, optimization problems associated to DG planning are, in general, non–linear, non– convex and combinatorial in nature. Non–linearity can be given by the power flow equality constraints and/or objectives functions involving power losses. The non–convex and combinatorial characteristics are mainly due to the decision variables representing discrete (integer) locations, number of units and type of technology of DG. Non–convex mixed–integer non–linear problems (MINLP) are difficult to solve by conventional mathematical models, with multiple local optimaand at least non–deterministic polynomial–time hard (NP–hard) computational complexity [7, 20]. This calls for alternative methods of solutions, like heuristic optimization techniques (HO) belonging to the class of evolutionary algorithms (EAs), which have been proposed as a most effective way of solution. These methods are suited to cope straightforwardly with non–convex combinatorial problems, discontinuous feasible spaces, non–linear and non–differentiable objective functions [20, 42].

1.3 Research objectives and original contributions

The objectives of the present thesis focus on the design and development of a modeling, simulation and optimization framework for the integration of renewable DG into electric power networks. Specifically, the problem considered is the selection of the technology, size and location ofmultiple DG units, under technical, operational and economic constraints. The following key research questions are addressed:

(i) Representation and treatment of the uncertain operational inputs, like the availability of diverse primary renewable energy sources, bulk–power supply, power demands and occurrence of components failures. 1 Introduction 11

(ii) Propagation of uncertainties onto the model of system operational response and control of the associated risk.

(iii) Computational efforts resulting from the combinatorial optimization problem associated to renewable DG integration.

In answer to the key research questions formulated, the main original contributions of this thesis work are:

(b) Integration of two indicators to measure and control uncertainty and risk, namely conditional value–at–risk (CVaR) and conditional value–at–risk deviation (DCVaR), respectively.

(c) With respect to the optimal technology selection, size and location of the renewable DG units, two distinct multi–objective optimization (MOO) strategies have been implemented by heuristic optimization (HO) search engines, in which the MCS–OPF model is nested to assess the performance of each DG–integrated network proposed along the evolutionary searching process:

In the first approach, the fast non–dominated sorting genetic algorithm (NSGA–II) is ◦ used for simultaneous minimization of the expected values of CG and ENS (ECG and EENS, respectively) combined with their respective CVaR(CG) and CVaR(ENS) values. The second approach performs a MOO differential evolution (DE) search to minimize ◦ concurrently ECG and its associated deviation DCVaR(CG).

(d) To cope with the large computational efforts required by the developed MOO frameworks with nested MCS–OPF, an original technique is introduced which embeds hierarchical clustering analysis (HCA) within a DE search engine. The technique identifies, in a controlled manner, groups of similar individuals (DG plans) in the DE population and, then, evaluates ECG performing MCS–OPF on selected representative individuals of the groups only, thus reducing the number of objective function evaluations in each iteration of the DE evolution loop. 12 1 Introduction

1.4 Thesis structure

This thesis manuscript is divided into two parts. The first part is composed by six Chapters which introduce the readers to the research context, the motivations and objectives, and present in detail the methodological developments performed to address the specific problem of optimal DG planning under uncertainty. In particular, Chapter2 provides the basics of the modeling of a DG–integrated electric power network, focusing on the system representation, the different models use to treat the uncertain operational inputs considered, the construction of the MCS–OPF computational model and the respective evaluation of the system performance based on economic, and uncertainty and risk measures. In Chapter3, the formulation of diverse optimization strategies to address the optimal DG planning problem and the HO with nested MCS–OPF frameworks developed are presented. Chapter4 briefly summarizes the use of clustering techniques in HO to increase the computational performance and, then, gives a complete description of the development of the DE search engine with embedded HCA. In Chapter5, examples of applications of the proposed frameworks are illustrated, with reference to an adaptation of the IEEE 13 bus distribution test feeder [2] and a realistic setting of the IEEE 30 bus sub–transmission and distribution test system of literature [11]. Finally, Chapter6 draws the conclusions of the thesis work carried out and proposes some opportunities for future research advancements. Figure 1.4 shows a diagram with the overall vision of the thesis structure.

The second part, includes the collection of papers published or under revision, which are the result of the research work here performed, and that the reader can refer to for further details. Paper (i) presents a direct risk–based simulation and MOO framework for the integration of renewable DG and storage based on NSGA–II, introducing the CVaR to find optimal DG plans, trading–off expected performance and risk. Paper (ii) addresses the challenge of reducing the computational efforts required to implement HO search engines with nested MCS–OPF: the technique developed integrates HCA and DE for optimal integration of renewable DG and, by defining control parameters, adapts itself along the evolutionary search determining whether it is convenient to perform clustering of the decision variables or not and at what scale to, then, reduce the number of objective function evaluations. Paper (iii) introduces a MOO framework for risk–controlled integration of renewable DG into electric power systems, which is based on DE search and MCS–OPF. This framework measures uncertainty in the system performance by the use of DCVaR that, due to its axiomatic relation to the CVaR, allows the conjoint control of risk. 1 Introduction 13

2 Renewable DG–integrated electric power network modeling 2.1 System representation 2.3 Stochastic operation modeling 2.1.1 Components classification 2.3.1 Non–sequential MCS 2.1.2 Network topology 2.3.2 Optimal power flow 2.2 Uncertain operational inputs • DC–OPF formulation 2.2.1 Power demands and energy price 2.4 Performance evaluation 2.2.2 Bulk–power supply 2.4.1 Global cost 2.2.3 Solar photovoltaic generation 2.4.2 Energy not supplied 2.2.4 Wind turbines generation 2.4.3 Uncertainty and risk measurement 2.2.5 Electrical vehicles • CVaR 2.2.6 Storage devices • DCVaR 2.2.7 Components availability states

3 Renewable DG–integrated electric power network planning 3.1 Optimal DG technologies selection, sizing and allocation 3.1.1 Optimization strategies formulation • (MO) weighted ECG, CVaR(CG)& EENS, CVaR(ENS) minimization • (MO) ECG & DCVaR(CG) minimization • (SO) ECG minimization 3.2 HO & MCS–simulation frameworks 4 Computational Challenge 3.2.1 NSGA II–based approach 4.1 Clustering in HO 3.2.2 DE–based approach 4.2 Hierarchical clustering & DE

5 Applications IEEE 13 bus test feeder 5.1 A risk–based MOO and MCS–simulation framework for the integration of renewable DG and storage devices 5.1.2 CVaR: expected performance and risk trade–off 5.1.3 Optimal DG–integrated network plans 5.1.4 Brief summary 5.2 HCDE for optimal integration of renewable DG reducing computational efforts 5.2.2 Quantification of the benefits of the HCDE 5.2.3 Identification of time complexity conditions and limits 5.2.4 Optimal DG–integrated network plans 5.2.5 Brief summary IEEE 30 bus test feeder 5.3 A MOO and MCS–simulation framework for risk–controlled integration of DG 5.3.2 DCVaR: expected performance, uncertainty and risk trade–off 5.3.3 Optimal DG–integrated network plans 5.3.4 Brief summary

6 Conclusions 6.1 Original contributions 6.2 Future work

Figure 1.4 Thesis structure diagram

2 Renewable DG–integrated electric power network modeling

In this Chapter, the MCS–OPF is presented, including representation of the DG–integrated network, the modeling of the different uncertain operational inputs considered, the process of generating the random operational scenarios and the OPF problem formulation. The definition of the performance evaluation function, based on economics, reliability of power supply and uncertainty–risk measurement, is introduced as well.

2.1 System representation

2.1.1 Components classification

The starting point for modeling a DG–integrated network is the definition of the type of components involved. For example in Figure 2.1 which illustrates the IEEE 30 bus sub–transmission and distribution test system, three main classes of components can be identified: power generators, T&D lines and loads.

The power generator class G considers the subclasses containing all the different types of conventional bulk power suppliers and renewable technologies to be integrated, denoted by MG and RG, respectively. In this study, four different types of DG technologies are considered: solar photovoltaics, wind turbines, electric vehicles and storage devices. The nomenclature denoting the classes, subclasses and types of components is summarized as follows:

G: Class of power generation components, G = MG RG. ◦ ∪ ◃ MG: Subclass of bulk–power generation components.

◃ RG: Subclass of renewable DG components, RG = PV W EV ST. ∪ ∪ ∪ PV : Solar photovoltaics. W: Wind turbines. EV : Electrical vehicles. 16 2 Renewable DG–integrated electric power network modeling

ST: Storage devices. TD: Transmission and distribution lines components. ◦ L: Power demand components. ◦ 30 29 THREE WINDING TRANSFORMER COVERDALE EQUIVALENTS 27 28 HANCOCK ROANOKE 12 10 C 9 C 13 4 11 26 6 24 25 23 33 (kV) 15 132 (kV) 14 18 Load 21 G Generator 19 22 C Synchronous condenser G 16 20 node i = 1 GLEN LYN 12 17 13 10 3 C HANCOCK 8 4 C 11 KUMIS ROANOKE 9 6 REUSENS 7 node i’ = 2 BLAINE CLAYTOR G T&D line (i,i’) = (2,5) C FIELDALE 5

Figure 2.1 IEEE 30 bus sub–transmission and distribution test system diagram

2.1.2 Network topology

The network topology is described as a graph, i.e., a set of nodes and the various connections that link them. The nodes represent spatial points at which generation components (MG and RG) and loads are located or can be allocated, whereas the connections between nodes are the transmission and distribution lines. A single node and the set of all nodes are indicated by the index i and N = i : i 1, 2, . . . , n , respectively, where n – total number of nodes in the network. { ∈ { }} Consequently, the links connecting nodes define the set of transmission and distribution linesas follows: Y = (i, i') : nodes i and i' are connected, i, i' N (2.1) { ∀ ∈ } 2 Renewable DG–integrated electric power network modeling 17

Assuming stationary operational conditions, the network performance is considered to be dictated by the locations and magnitudes of the power available in each generation unit, the loads and the technical limits of the T&D lines. To indicate the location and capacity size of the different types of generation units present in the network, the matrix Q, i N is defined by the following expression: ∀ ∈

(m + r) types of power generators G

m types of bulk–power suppliers MG r types of renewable technologies RG

G G G1 j Gm G1+m j+m Gr+m ··· ··· ··· ···

q1,1 q1,j q1,m q1,1+m q1,j+m q1,r+m . ···. . ···. . . ···. . ···...... Q q q q q q q  = i,1 i,j i,m i,1+m i,j+m i,r+m (2.2)  . ···.. . ···.. . . ···.. . ···.. .   ......  q q q q q q   n,1 n,j n,m n,1+m n,j+m n,r+m  ··· ··· ··· ···  QM QR where, QM , QR – matrices of location and capacity size of bulk–power suppliers (MG) and renewable power generators (RG), respectively, j – power generator type index, m – total number of types of bulk–power suppliers, r – total number of types of renewable technologies and [Q]i,j = qi,j – non–negative integer that specifies the number of units of the power generator type j allocated at node i:

q' Z∗ if q' units of Gj are allocated at node i, Gj G1,..., Gm+r qi,j = ∈ ∀ ∈ { } (2.3)  0 otherwise 

Then, a proposed plan of DG integration is represented by the matrix QR and its concatenation to the fixed locations and sizes of the bulk–power generators, already present in the existing network, M M R Q results in the complete static deployment of all power generator components Q = [Q Q ]. | Therefore, the complete representation of a DG integrated is given by the pair ([Q], Y ). Any physical component, G or TD, is assumed to be affected by the stochastic occurrence{ of failures,} conditioning dynamically the functionality of power generators and the lines through which the power flows. Furthermore, the magnitude of power available in each generation unit issubject to the intrinsic uncertain behavior of the corresponding primary energy source and, under the assumption that generators act as price–takers, the economic conditions depend on the variability of the power demands [4, 52]. The aforementioned uncertain conditions significantly affect the operation of a given DG– integrated network, therefore, modeling the diverse sources of variability becomes essential to 18 2 Renewable DG–integrated electric power network modeling emulate the operational response of the system for a large representative combination of possible scenarios and, ultimately, be able to assess its probabilistic performance relative to preset target functions.

2.2 Uncertain operational inputs

In principle, analytical methods are preferable to Monte Carlo simulation (MCS) because of the possibility of achieving accurate solutions; however, their application usually requires simplifications in the modeling which may lead to unrealistic results. An example of this are analytical solutions for optimal DG planning that do not take into account uncertainty or intermittency in power generation and/or load profiles, and networks of low dimensionality [74]. On the other hand, MCS allows a more realistic modeling, because the operation of the network is not analytically solved but simulated, and the overall performance indicators are statistically estimated from virtual operational scenarios realizations [75]. MCS has been found quite suitable for the analysis of electric power networks with multiple sources of uncertainty, e.g., power generation, loads, component failures or degradation processes, etc. [37, 42, 43, 62, 63, 76], but at the expense of requiring more computational resources.

In the present thesis work, we adopt a non–sequential MCS, based on latin hypercube sampling (LHS) [77], to emulate the operation of the DG–integrated network, considering the realizations of uncertain operational variables as independent on previous realizations, so as to seize the advantages of MCS without overly increasing the computational efforts.

2.2.1 Power demands and energy price

Overall power demand profile in an electric power network, as well as single nodal load profiles, can be inferred from historical data as daily load curves, in which to each hour of the day corresponds one specific level of load [10, 52]. In addition, power demands can be considered uncertain following normal distributions [42, 76]. Here, both models are integrated, adopting normally distributed nodal load profiles for which their respective mean and standard deviation parameters vary depending on the hour of the day t D = 1,..., 24 , as shown in Figure 2.2. Then, the power demand at node i is modeled as: ∈ { }

φ(ξ(Li,t , µi,t , σi,t )) L , µ , σ 0 σ Z µ , σ i,t i,t i,t fi,t (Li,t µi,t , σi,t ) = i,t ( i,t i,t ) ∀ ≥ (2.4) |   0 otherwise L µ µ  i,t i,t i,t ξ(Li,t , µi,t , σi,t ) = − ; Z(µi,t , σi,t ) = 1 Φ (2.4a) σi,t − σi,t 2 Renewable DG–integrated electric power network modeling 19

μi,t ) σi,t W

M ∙∙∙ (

t f (L ) , i,t i,t i L

0 t (h) 23

Figure 2.2 Example of a nodal daily load profile, hourly normally distributed

where, fi,t (Li,t µi,t , σi,t ) – truncated Normal probability density function, Li,t (MW) – power demand | at node i at hour of the day t, µi,t , σi,t (MW) – Normal distribution mean and standard deviation, respectively, φ, Φ – standard Normal (pdf) and its cumulative distribution function (cdf), respectively.

The power generators in the network are assumed to be price–takers, for which the value of the energy price is correlated with the aggregated power demand. As an intermediate approximation of existing studies in [4, 52, 53] (Figure 2.3), the proportional correlation used in this study can be expressed as:

0.75

EP t 0.5 EPmax

0.25

0 0 0.25 0.5 0.75 1

Li,t ¦ L iN maxi

Figure 2.3 Proportional correlation energy price vs aggregated load

L L EP L EP , L EP i,t 0.38 i,t 1.38 (2.5) t ( i,t max maxi ) = max L L + | i N maxi − i N maxi X∈ X∈ where, EPt ($/MWh)– energy price at hour of the day t, EPmax ($/MWh)– maximum value of energy price and L MW– maximum value of power load at node i. maxi 20 2 Renewable DG–integrated electric power network modeling

2.2.2 Bulk–power supply

Bulk–power stands for the power supply coming from conventional power plants (MG) already existing in the network. These are rather stable and are connected to the network at sub–transmission or distribution transformers to provide the voltage level of the customers. The stochastic behavior of the available power in these sources is represented following normal distributions [20, 78], with small standard deviation and truncated by the maximum capacity of generation.

φ(ξ(Pj, µj, σj)) P 0, P , µ , σ 0 j [ max j ] j j f P µ , σ , P σj Z(µj, σj) (2.6) j( j j j max j ) = ∀ ∈ ≥ |   0 otherwise P µ P µ µ j j max j j j ξ(Pj, µj, σj) = − ; Z(µj, σj) = Φ( − ) Φ( ) (2.6a) σj σj − σj where j j : G MG , f P µ , σ , P – truncated Normal pdf, P , P (MW) – available j j( j j j max j ) j max j ∀ ∈ { ∈ } | bulk power and maximum capacity of the MG generator type j, respectively, µj, σj (MW) – Normal distribution mean and standard deviation.

2.2.3 Solar photovoltaic generation

PV technologies (PV) converts solar irradiance into electric power through a set of solar cells configured. Commonly, solar irradiance uncertain behavior has been modeled using probabilistic distributions, obtained from long term weather historical data of a particular geographical area. The Beta distribution function has been found particularly suitable to model hourly solar irradiance [10, 79]. The intermittency in the solar irradiation is taken into account defining a daylight interval between 07.00 and 21.00 hours, setting a positive value of solar irradiation H if the value t of the hour of the day is in the subset of DL = 7,..., 21 of D, otherwise, the value of solar irradiance is assumed equal to 0 given that t is in the{ night interval.} Thus, the Beta distribution function is adjusted considering the probability pL = P(t t DL) that t falls in the daylight interval: | ∈ α 1 ( i ) (βi 1) Hi − (1 Hi) − − Hi H , 1 , αi, βi > 0 1 p B α , β [ i∗ ] fi(Hi αi, βi, pL, Hi∗) = ( L) ( i i) ∀ ∈ (2.7) |  −  0 otherwise

 Γ (αi)Γ (βi) B(αi, βi) = (2.7a) Γ (αi + βi) where fi(Hi αi, βi, pL, Hi∗) – adjusted Beta probability density function, Hi – solar irradiance at node | i, αi, βi – shape parameters of the corresponding Beta distribution at node i, Hi∗ – pL percentile of the non–adjusted Beta pdf fi(Hi αi, βi) and Γ – Gamma function. | 2 Renewable DG–integrated electric power network modeling 21

Besides the dependence on solar irradiance, the available power output is determined by the technical characteristics of the PV cells and the ambient temperature on site. Then, the available power output provided by nc solar cells can be obtained from the following equations:

P' H if 0 P' H P i,j( i) i,j( i) max j Pi,j(Hi) = ≤ ≤ (2.8)  Pmax if Pmax < P'i,j(Hi)  j j P' H n FF V H I H 10 6 (2.8a) i,j( i) = c j ( i) ( i) − × T H T H T 20 /0.8 (2.8b) C ( i) = Ai + i( N oj + ) I T H I k T 25 (2.8c) ( C ) = i( SCj + I j ( C )) − V T V k T (2.8d) ( C ) = OCj + Vj C

VMPP IMPP FF j j (2.8e) j = V I OCj SCj where j j : G PV , P (MW) – power output at node i, P (MW) – maximum power j i,j max j generation∀ ∈ capacity, { ∈FF –} fill factor, T ( C) – ambient temperature at node i, T ( C) – nominal j Ai ◦ N oj ◦ cell operation temperature, I (A) – short circuit current, k (mA C) – current temperature SCj I j /◦ coefficient, V (V) – open circuit voltage, k (mV C) – voltage temperature coefficient, and OCj Vj /◦ V (V), I (A) – voltage and current at maximum power, respectively. MPPj MPPj

2.2.4 Wind turbines generation

Wind power generation (W) is obtained from turbine–alternator devices that transform the kinetic energy of the wind into electric power. The stochastic behavior of the wind speed is commonly represented through probability distribution functions. The Weibull distribution has been widely used to model the randomness of the wind speed in various conditions [10, 20, 34, 73, 79, 80]:

(β∗i 1) β∗i β∗i Ui − Ui exp Ui 0, α , β > 0 f U α , β α α α ∗i ∗i (2.9) i( i ∗i ∗i ) =  ∗i ∗i − ∗i ∀ ≥ |  0 otherwise

 where, fi(Ui α∗i , β∗i ) – Weibull probability density funtion, Ui (m/s) – wind speed at node i and α∗i , | β∗i – scale and shape parameters of the Weibull distribution function at node i, respectively. 22 2 Renewable DG–integrated electric power network modeling

Similarly to PV type of technologies, the uncertainty associated to the wind speed and the technical features of a specific type of wind turbine characterize its power output function:

U U i CI j − P if U U < U U U R j CI j i Aj Aj CI j ≤  − Pi,j(Ui) = P if U U U (2.10)  R j Aj i COj  ≤ ≤ 0 otherwise   where j j : G W , P (MW) – rated power and U , U , U (m s) – cut–in, average and j R j CI j Aj COj / cut–out∀ wind∈ { speeds,∈ respectively.}

2.2.5 Electrical vehicles

In this study, electric vehicles (EVs) are considered as battery electric vehicles with three possible operating states ρ:( 1) charging, (0) disconnected and (1) discharging [32]. When in charging state, an EV act as a− power demand, whereas in discharging it injects power into the network. EVs operation is modeled considering them as ‘block groups’, i.e., EVs sharing similar operational patterns are aggregated into a single block. In fact, it has been observed that EVs present nearly stable daily usage schedules, in addition, modeling them as ‘block groups’ contributes to the need of avoiding the combinatorial explosion of the model [5]. The power output of one block of EVs is formulated by assigning residence time intervals tρ to each possible operating state and associating them with the percentage of trips that the vehicles perform by hour of a day [32]. This allows approximating the hourly probability distribution of the operating states per day, as shown Figure 2.4. Then, the random determination of the operating state

ρj,t of a block of EVs of type j, given a specific hour of the day t, is sampled from the corresponding probability f j,t (ρj,t ) associated to the occurrence of each state.

1.001.00  chargingp,j ,t ρ1 j ,t  0 0.750.75 disconnectedp,j ,t ρ0 j ,t   dischargingp,j ,t ρ1 j ,t  ) t , j

r 0.500.50 ( t , j f 0.250.25

0.000.00 00 11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 t (h)

Figure 2.4 Example of hourly probability distribution of EV operating states per day 2 Renewable DG–integrated electric power network modeling 23

p−j,t if ρj,t = 1 − f ρ p , p0 , p+ 0 (2.11) j,t ( j,t −j,t j,t j,t ) =  pj,t if ρj,t = 0 |   + pj,t if ρj,t = 1  0 +  where j j : Gj EV , p−j,t , pj,t , pj,t – hourly per day probability distribution of a block of EVs ∀ ∈ { ∈ } type j for the respective operating states, ρj,t = 1 (charging), ρj,t = 0 (disconnected), ρj,t = 1 (discharging). −

Accordingly, the power output for a single EV is calculated using the expression (2.12) below:

P ρ ρ P ∆t 0, tρ (2.12) j,t ( j,t ) = j,t R j [ ] ∀ ∈ where j j : G EV , tρ – residence time interval for operating state ρ (h), P (MW) – rated j R j power∀ (MW).∈ { ∈ }

2.2.6 Storage devices

Analogously to the EV case, storage devices (ST) are treated as batteries. In reality, these present two main operating states, charging and discharging [81]. However, in accordance to the non–sequential characteristic assumed for the MCS model, for this study the level of charge in the batteries is randomized and the state of discharging is the only one that takes place, making them independent J on previous states of charge. The discharging time interval t j is assigned according to the relation between the batteries rated power of type j, their energy density J and the random level of charge Sj J they present. For this, the discharging action is carried out at a rate equal to the rated power. Then, the power output per unit of mass of active chemical in the battery M is estimated as follows: Tj

1 J 0, J M J M Sj Tj f J , J , M Sj Tj ∀ ∈ (2.13) j( Sj Tj ) = |   0 otherwise P J P ∆t 0, t J (2.14) j( ) = R j [ j ] ∀ ∈ J t J (2.14a) = P R j where j j : G ST , f J , J , M – Uniform probability density function, J (MJ) – level of j j( Sj Tj ) charge∀ in the∈ { storage∈ device,} J| (MJ kg) – specific energy of the active chemical, M (kg) – mass Sj / Tj of active chemical, P (MW) – rated power, t J (h) – upper bound of discharging time interval. R j j 24 2 Renewable DG–integrated electric power network modeling

2.2.7 Components availability state

Consistently with the non–sequential nature of the proposed MCS simulation, the availability states of the physical components in the network, generators (G) and T&D lines (TD), are direclty modeled by two–state stationary Markov chains [5, 37], defining two possible operating states: η = 0 if the corresponding component is non functional (failure) and η = 1 if the component is available to operate, i.e., be able to generate, transmit or distribute power accordingly to the class of component. Then, the discrete stationary distribution of operating states can be expressed as follows:

λ / λ λ η 0 Fk ( Fk + Rk ) k = fk(ηk λF , λR ) = (2.15) k k  | λR /(λF + λR ) ηk = 1  k k k where k k : G G k/k i, i' Y , η – operating state of component k and λ (n h) k = ( ) k Fk / and λ ∀ (n∈h) {{ – failure∈ and} ∪ { repair rates,∈ respectively.}} Rk /

2.3 Stochastic operation modeling

2.3.1 Non–sequential Monte Carlo simulation

M R For a given DG–integrated network plan, denoted by the pair ([Q], Y ), recalling that Q = [Q Q ], and that represents the locations and number of units of the{ different} power generators| and the T&D lines, each uncertain variable is randomly sampled several times by LHS [77] and the ∆ inverse transform method [75], for the realization of NS operational scenarios of duration t . For practicality, we define NS as multiple of 24, so each hour of the day t has the same number of realizations NS/24. The set ω contains all the sampled variables which constitute an operational scenario which, conjointly to the pair ([Q], Y ), set the stage for evaluating the response of the network in terms of available power usage,{ } power demand satisfaction and the involved economics. Ω is defined as the set of all the NS realizations of ω, the respective notation is given by Equations (2.16) and (2.16).

Figure 2.5 shows schematically the sampling process of the uncertain variables from the models presented in the preeciding section.

ω t , L , EP , P , G MG , η , η , H , U , ρ , J (2.16) s = s i,ts ts i,j,s j (i,i'),s i,j,s i,s i,s j,t,s s ∀ ∈ Ω = ωs : s 1, 2, . . . , NS (2.17) { ∈ { }} 2 Renewable DG–integrated electric power network modeling 25

(2.4) L F 1 u 0, 1 µ , σ (2.5) EP L EP , L i,ts = i−,ts ( ( )i i,ts i,ts ) i N ts ( i,ts max maxi ) { | }∀ ∈ | 1 (2.15) η i,i' ,s F u 0, 1 i,i' λF , λR i,i' Y ( ) = (−i,i')( ( )( ) (i,i') (i,i') ( ) { | }∀ ∈ η F 1 0, 1 λ , λ (2.15) i,j,s = j− (u( )i,j Fj R j [Q]i,j =0 { | }∀ ̸ P F 1 0, 1 µ , σ , P (2.6) i,j,s = j− (u( )i,j j j max j ) [Q]i,j ,ηi,j,s=0,Gj MG { | }∀ ̸ ∈ H F 1 0, 1 α , β , p , H (2.7) i,s = i− (u( )i i i L i∗) [Q]i,j ,ηi,j,s=0,Gj PV (2.8) { | }∀ ̸ ∈ U F 1 0, 1 α , β (2.9) i,s = i− (u( )i ∗i ∗i ) [Q]i,j ,ηi,j,s=0,Gj W (2.10) { | }∀ ̸ ∈ P i,j,s [Q]i,j =0,Gj G (2.11) ρ F 1 u 0, 1 p , p0 , p+ (2.12) { }∀ ̸ ∈ j,ts = j−,ts ( ( )i,j −j,ts j,ts j,ts ) [Q]i,j ,ηi,j,s=0,j EV { | }∀ ̸ ∈ J F 1 0, 1 , J , M (2.13) i,j,s = j− (u( )i,j Sj Tj ) [Q]i,j ,ηi,j,s=0,Gj ST (2.14) { | }∀ ̸ ∈ Figure 2.5 Sampling process

2.3.2 Optimal power flow

Power flow analysis is performed by the DC approximation [82] which takes into account only the active power, neglecting power losses, and assumes a flat voltage profile throughout the network. This allows transforming to linear the classic non–linear power flow equality constraints, gaining simplicity and computational tractability. DC power flow is often used in techno–economic analysis of electric power systems, more frequently in transmission [82, 83] but also in distribution networks [83]. The generic DC power flow equations are:

Pi = Sre f B(i,i')(δi δi') i N, (i, i') Y (2.18) i' N − ∀ ∈ ∈ X∈ P L P 0 i N (2.19) Gi i i = i N − − ∀ ∈ X∈ where, Sre f (MV A) – reference apparent power in the network, Pi (MW) – active power leaving or entering node i, B (p.u.) – susceptance of the T&D line i, i' , δ – voltage angle at node i, P (i,i') ( ) i Gi (MW) – active power injected or generated at node i and Li (MW) – load at node i. The assumptions are:

• the difference between voltage angles is small, i.e., sin(∆δ) ∆δ, cos(∆δ) 1. ≈ ≈ • the resistance of the T&D lines are neglected, i.e., R X , thus, that power losses are neglected as well. ≪

• the voltage profile is flat, constant V , set to 1 (p.u.) 26 2 Renewable DG–integrated electric power network modeling

DC–OPF formulation

DC optimal power flow analysis (OPF) is run, taking in input the configuration ([Q], Y ) and each { } operating scenario ωs Ω and aiming at the minimization of the aggregated operating cost CO. The quantity CO has been∈ defined in two different manners: (i) G CO considers solely the operating costs concerning generation of power, (ii) GTDCO accounts for the aggregation of the operating costs of generation, transmission and distribution and load shedding, including revenues per MW h sold. The present formulation of the power flow problem is:

MCS–OPF([Q], Y , ωs ): { } ω min COs (2.20) PUs ,∆δs,LSs (i) GCOω t∆ COv P (2.20i) s = j Ui,j,s i N Gj G X∈ X∈ (ii)

GTDCOω COv EP P C EP LS s = ( j ts ) Ui,j,s + ( LS + ts ) i,s+ i N Gj G − i N X∈ X∈ X∈ (2.20ii) Sre f COv(i,i') B(i,i')(δi,s δi',s) (i,i') Y | − | X∈ s. t.

L P S η B δ δ LS 0 (2.21) i,ts Ui,j,s re f (i,i'),s (i,i')( i,s i',s) i,s = − Gj G − i' N − − X∈ X∈ 0 P η Q P (2.22) Ui,j,s i,j,s[ ]i,j i,j,s ≤ ≤ S B δ δ P (2.23) re f (i,i')( i,s i',s) max i,i' | − | ≤ ( ) G ω GTD ω where t D, s 1, 2,..., NS , COs ($) – aggregated operating cost of generation, COs ($) ∀ ∈ ∈ { } – aggregated operating cost of generation, transmission and distribution, and load shedding, COvj ($ MWh) – variable operating cost of the power generator j, EP ($ MWh) – energy price at hour t, / ts / P (MW) – used power from the generator type j located at node i, COv ($ MWh) – variable Ui,j,s (i,i') / operating cost of the T&D line i, i' , C ($ MWh) – load shedding cost and P (MW) – power ( ) LS / max(i,i') rating of the T&D line (i, i'). The load shedding LSi,s (MW) at node i is defined as the amount of load disconnected to alleviate congestion in the T&D lines and/ or balance the demand of power with the available power supply.

The meaning of each constraint is:

(2.21): power balance at node i. ◦ 2 Renewable DG–integrated electric power network modeling 27

(2.22): bounds of the power generation units. ◦ (2.23): technical limits of the T&D lines (power rating). ◦

The OPF problem is solved for each operational scenario ωs, giving in output the corresponding ω Ω ω ω ω values of minimum COs . The set CO = CO1 , CO2 ,..., CONS is, then, considered as a sample of realizations of the probability density function{ of CO. }

2.4 Performance evaluation

M R The proposed renewable DG–integrated network solutions Q = [Q Q ] are evaluated with respect to performance indicators regarding economics and reliability of power| supply. Specifically, the expected values of performance and the associated uncertainty–risk measures are considered as targets.

2.4.1 Global cost

The economic performance of the network, given a DG integration plan, is evaluated with respect to a global cost function (CG). The quantity CG is composed by two terms: the outcome operating cost of the MCS–OPF described in the previous section, COΩ, and the fixed investment and operating cost, R CI j + CO f j, associated to the renewable part of the proposed DG plan Q , i.e., j j : Gj RG . ∀ ∈ { ∈ } The fixed investment and operating cost CI j + CO f j ($) is prorated hourly over the lifetime of the project t H . Thus, the global cost function for the set of operational scenarios Ω is given by the following equations differentiated according the two distinct definitions of COΩ (Equations (2.20i) and (2.20ii)):

(i)

∆ G Ω G Ω t R ∆ Ω Ω CG = CO + CI j + CO f j [Q ]i,j t (inc + EP )P (2.24i) t H Ui,j i N Gj RG − X∈ X∈ (ii) 1 GTDCGΩ GTDCOΩ CI CO f QR (2.24ii) = + t H j + j [ ]i,j i N Gj RG X∈ X∈ where, EPΩ – sample of NS realizations of EP , inc ($ MWh) – incentive for power generation ts / from DG sources and PΩ – sample of NS realizations of P . Ui,j Ui,j,s Analogously to COΩ definition, CGΩ represents a sample of realizations of the probability density function of CG and performance indicators of interest can be obtained, relative to expected performance, uncertainty and risk. 28 2 Renewable DG–integrated electric power network modeling

2.4.2 Energy not supplied

The energy not supplied (ENS) is a common index for reliability of power supply evaluation [3, 10, 20, 34, 45, 84–87]. In the present work, its value is obtained directly from the OPF outputs in the form of the aggregation of all nodal load sheddings per scenario ωs Ω: ∈ Ω ∆ Ω ENS = t LSi (2.25) i N X∈ Ω ω Ω where, ENS – sample of NS realizations of ENSs (MW h) and LSi – sample of NS realizations of LSi,s (MW h).

2.4.3 Uncertainty and risk measurement

The proposed framework introduces the CVaR and CVaR deviation (DCVaR) [66] to measure, respectively, the risk and uncertainty in the performance functions of interest: CG and ENS. The quantity DCVaR is a functional of the CVaR [88], which is a coherent risk measure broadly used in financial portfolio optimization and has been extended to engineering applications, including electric power systems analysis and, in particular, DG planning [61, 63, 64, 69–72]. The definitions and properties of CVaR and DCVaR for continuous and discrete general return (loss) functions are given in detail in [66, 88]. Here, only a graphical, but comprehensive view to understand the CVaR and DCVaR definitions is presented in Figure 2.6.

E(x)

DCVaR(x) = CVaR(x - E(x)) frequency

VaR(x)

CVaR(x)

%ile loss x

Figure 2.6 Graphic representation of VaRα(x), CVaRα(x) and DCVaRα(x); x = loss

For a discrete approximation of the probability function of the loss x, given a confidence level or α-percentile, the value–at–risk VaRα(x) represents the smallest value of loss for which the probability that the loss does not exceed that threshold value is greater than or equal to α, whereas 2 Renewable DG–integrated electric power network modeling 29

CVaRα(x) is the expected value of loss given that the loss is greater than or equal to the VaRα(x). Thus, CVaRα(x) provides a quantitative indication of the extent of the probability of occurrence of extreme non–desirable or risky scenarios of loss. The quantities VaRα(x) and CVaRα(x) can be expressed by the following equations:

VaRα(x) = inf z : Fx (z) > α (2.26) { }

CVaRα(x) = E(x/x VaRα(x)) (2.27) ≥ where, Fx – cumulative distribution function of the loss x.

With regards to DCVaRα(x), this is a non symmetric deviation measure, as it accounts for the uncertainty associated to the loss exceeding its expected value. It is defined taking into account some important properties of the standard deviation [66], and is formulated as:

DCVaRα(x) = CVaRα(x E(x)) (2.28) − Furthermore, being a coherent risk measure, CVaR is a strictly expectation–bounded risk measure and it can be proved that a one–to–one relation exists with its corresponding deviation measure DCVaR [66]: CVaRα(x) = E(x) + DCVaRα(x) (2.29)

In the present framework, a specific configuration of the DG–integrated network ([Q], Y ) can R { } be considered as a generation portfolio, in which the renewable part [Q ] of [Q] is the decision matrix. Ω Ω The corresponding assessed CG and ENS , obtained from the output MCS–OPF([Q],{Y},Ω), can be Ω Ω translated into the probability functions of loss; then, the quantities CVaRα(CG ), DCVaRα(CG ) Ω Ω and CVaRα(ENS ), DCVaRα(ENS ) represent the level of risk and uncertainty associated to the R Ω solution [Q ] with an expected global cost ECG = E(CG ) and expected energy not supplied Ω EENS = E(ENS ), respectively.

3 Renewable DG–integrated electric power network planning

This Chapter presents the distinct optimization frameworks considered to address the optimal integration of DG in terms of selection of technology, sizing and allocation of renewable DG units. R M R The corresponding decision matrix [Q ] is contained in the matrix Q = [Q Q ] that stores the number and location of each type of power generator in the network. |

Three distinct optimization strategies are formulated, defining correspondingly three different frameworks (FWs). Framework number 1 (FW1) focuses on controlling the risk associated to the expected performance of the DG plans, ECG and EENS, by measuring their respective CVaR values,

CVaRα(CG) and CVaRα(ENS). FW2, aims at the control of uncertainty with respect to the ECG performance by targeting in conjunction the associated DCVaRα(CG). Whereas FW3, presents a single objective strategy, aiming solely ECG to address the challenge of reducing the computational efforts required to implement heuristic optimization (HO) search engines with nested MCS–OPF and it is treated in details in the next Chapter4.

3.1 Optimal DG technologies selection, sizing and allocation

3.1.1 Optimization strategies formulation

FW1: Multi–objective weighted ECG, CVaR(CG)& EENS, CVaR(ENS) minimization

The MOO problem consists in the concurrent minimization of the two objective functions measuring the CG and ENS. Specifically, their expected values and their CVaR values are combined, weighted by a factor β [0, 1], which allows modulating the expected performance of the DG–integrated network and its∈ associated risk. 32 3 Renewable DG–integrated electric power network planning

Considering a set of randomly generated scenarios Ω, the MOO problem is formulated as follows:

Ω Ω † min βE CG 1 β CVaRα CG (3.1) R ( ) + ( ) ( ) † Ω G Ω [Q ]i,j − CG = CG by (2.24i) min βE ENSΩ 1 β CVaR ENSΩ (3.2) R ( ) + ( ) α( ) [Q ]i,j − s.t.

[Q]i,j Z∗ (2.3) ∈ R CI j + CO f j [Q ]i,j BGT (3.3) i N Gj RG ≤ X∈ X∈ R [Q ]i,j τj Gj RG (3.4) i N ≤ ∀ ∈ X∈ MCS–OPF([Q], Y , Ω) (2.20)−(2.23) { } The meaning of each constraint is:

R (2.3): the decision variable [Q ]i,j is a non–negative integer number. ◦ (3.3): the total fixed investment and operating cost of the DG units must be less than orequal ◦ to the available budget BGT ($).

(3.4): the total number of DG units to allocate of each technology j must be less than or equal ◦ to the maximum number of units available τj to be integrated.

(2.20)−(2.23): the OPF equations of must be satisfied for all scenario ωs Ω. ◦ ∈ Constraint (3.4) can be translated into maximum allowed penetration factor PF DG of each max j DG technology j. Defining PF as ‘the output active power of total capacity of DG divided by the aggregated maximum nodal loads’ [23], constraint (3.4) can be rewritten as follows:

QR E P DG [ ]i,j ( j ) τ E P DG i N DG DG j ( j ) ∈ = PF PF = Gj RG (3.5) P L j max j L maxi ≤ maxi ∀ ∈ i N i N P∈ P∈ R DG where, [Q ]i,j – total number of units of DG technology j integrated in the network; E(Pj ) – expected power output of one unit of DG technology type j (MW); L – aggregated maximum P maxi nodal loads (MW). P 3 Renewable DG–integrated electric power network planning 33

FW2: Multi–objective ECG & DCVaR(CG) minimization

The practical aim of the MOO is the simultaneous minimization of representative indicators of the objective function CG, given by the expected value ECG and the associated uncertainty measure

DCVaRα(CG). The general MOO problem for all set of randomly generated operational scenarios Ω is formulated as follows:

min E CGΩ (3.6) R ( ) [Q ]i,j Ω ‡ min DCVaRα CG (3.7) R ( ) ‡ Ω GTD Ω [Q ]i,j CG = CG by (2.24ii) (3.8)

s.t.

[Q]i,j Z∗ (2.3) ∈ [Q]i,j PAV j PF DG (3.9) Lmax i N Gj RG i ≤ X∈ X∈ MCS–OPF([Q], Y , Ω) (2.20)−(2.23) { } The meaning of each constraint is the following:

(2.3): the decision variables [Q]i,j are non–negative integer numbers. ◦ (3.9): the ratio of total amount of average renewable power integrated in the network must ◦ be less than or equal to the penetration factor PF DG.

(2.20)−(2.23): the OPF equations of must be satisfied for all scenario ωs Ω. ◦ ∈

FW3: Single–objective ECG minimization

This single objective optimization strategy ought to find the optimal plan of integration of renewable R DG [Q ] by minimizing the expected value of global cost ECG. Considering a set of randomly generated scenarios Ω, the optimization problem is formulated as follows:

min E CGΩ ‡ (3.10) R ( ) ‡ Ω GTD Ω [Q ]i,j CG = CG by (2.24ii) 34 3 Renewable DG–integrated electric power network planning

s.t.

[Q]i,j Z∗ (2.3) ∈ R CI j + CO f j [Q ]i,j BGT (3.11) i N Gj RG ≤ X∈ X∈ R [Q ]i,j τj Gj RG (3.12) i N ≤ ∀ ∈ X∈ MCS–OPF([Q], Y , Ω) (2.20)−(2.23) { } The meaning of each constraint is:

R (2.3): the decision variable [Q ]i,j is a non–negative integer number. ◦ (3.11): the total investment and fixed operation and maintenance costs must be less thanor ◦ equal to the available budget BGT.

(3.12): the total number of renewable DG units of each technology j to be allocated must be ◦ less than or equal to the maximum number of units available for integration τj.

(2.20)−(2.23): the OPF equations of must be satisfied for all scenario ωs Ω. ◦ ∈

3.2 Heuristic optimization & MCS–OPF simulation frameworks

The MOO optimization problems are non–linear and non–convex, i.e., a non–convex mixed–integer non–linear problem or non–convex MINLP.Non–linearity is due to the fact that all the objective functions involved cannot be written in the canonical form of a linear program, i.e., C T X , where C is a vector of known coefficients and X the decision vector. In the present case, the decision matrix R [Q ] enters the MCS–OPF flow simulation to obtain the probability density functions of ENS and/or CG, then, the objective functions are formed by performance indicators of interests: expected values, CVaR and DCVaR, in correspondence to the framework applied. Thus, the operations carried out R on [Q ] through MCS–OPF, expected, CVaR and DCVaR values cannot not be represented as the T R R product C [Q ]. The problem is non–convex because the decision matrices [Q ] are integer–valued and, as it is known, the set of non–negative integers is non–convex.

Given the class of optimization problem in the proposed framework (non–convex MINLP), it is most likely to have multiple local minima. Moreover, the dimension of the distribution network can R lead to a combinatorial explosion of the feasible space of the decision matrices [Q ] [16, 20]. Heuristic optimization algorithms (HO) have emerged as the most effective search engines for combinatorial optimization problems and they can deal with non–differentiable objective functions, discontinuous feasible spaces and non–convex conditions [20, 42]. Some of the best known techniques are: particle swarm optimization (PSO) [9, 42, 56, 76, 89], differential evolution (DE) 3 Renewable DG–integrated electric power network planning 35

[51, 90, 91] and genetic algorithms (GA) [3, 16, 34, 37, 69]. In this view, the three non–convex MINLPs under uncertainties here proposed are solved by implementing HO search engines.

The MOO problem under uncertainties of FW1 is solved by the NSGA–II algorithm [92], in which the evaluation of the objective functions is performed by the developed MCS–OPF.In FW2, the MOO differential evolution (MOO–DE) algorithm is implemented, integrating a fast non–dominated sorting procedure and crowded–comparison operator [92] into the original single objective DE [93], and also evaluating the objective functions by MCS–OPF. FW3 performs the search for optimal solution by an original technique developed in this work, which integrates hierarchical clustering analysis (HCA) into the basic single objective differential evolution (DE) search, and is presented in Chapter4.

For MOO frameworks, FW1 and FW2, the extension to MOO entails the integration of Pareto optimality concepts. In general terms, solving a MOO problem of the form:

min f1(X ), f2(X ),..., fm(X ) X { } s.t. X Λ ∈ n with at least two conflicting objectives functions (fi : ) implies to find, within a set of acceptable solutions that belong to the non–empty feasibleℜ → region ℜ Λ n, the decision vectors ⊆ ℜ X Λ that satisfy the following [94]: ∈

X Λ/fi(X ) fi(X ′), i 1, 2, . . . , m and fi(X ) < fi(X ′) for at least one i ¬ ∈ ≤ ∀ ∈ { } ⇓ fi(X ) fi(X ′) i.e. X dominates X ′ ≺ The vector X is called a Pareto optimal solution and the Pareto front is defined as the set f (X ) n such that X is Pareto optimal solution. The general NSGA–II and MOO–DE algorithms{ are∈ describedℜ } next.

3.2.1 Non–dominated sorting genetic algorithm II–based approach

NSGA–II is one of the most efficient MOO evolutionary algorithms inHO [95]. It uses an elitist approach by applying a fast non–dominated sorting and crowding–distance comparison operator, while the search for non–dominated solutions is performed based on two main genetic operators, namely mutation and crossover [92]. The general NSGA–II algorithm is summarized as follows:

Initialization

Set the values of parameters: ◦ 36 3 Renewable DG–integrated electric power network planning

◃ NP: population size.

◃ NGmax : maximum number of generations.

◃ pC : crossover probability [0, 1]. ∈ ◃ pM : mutation probability [0, 1]. ∈ Form the initial population POP0, randomly generating NP decision matrices (individu- ◦ 0 0 0 0 als) X within the feasible space, POP = X1 ,..., Xk ,..., XNP 0 { 0 } 0 Evaluate the objective functions f1(Xk ),..., fNF (Xk ) for each individual Xk , where ◦ NF – number of objective functions. 0 Rank the individuals in POP , applying a fast non–dominated sorting procedure [92] ◦ with respect to the values of the objective functions and identify the non–dominated 0 0 0 0 fronts 1 ,..., ND , where 1 is the best front and ND the less good front. {F F } F F 0 0 Compute and assign the crowding–distance value (dC ) to each Xk individual in POP ◦ and sort, in ascending order with respect to dC , the individuals belonging to the same 0 non–domination–ranked group ℓ ℓ 1,...,ND . ∈{ } {F } 0 Apply binary tournament selection [92] to POP based on the crowding distance to ◦ generate an intermediate population POP0' of size NP. Reproduction ◃ Apply mutation and crossover operators to POP0', to create an offspring population OPOP0 of size NP. 0 0 ◃ Evaluate the objective functions for each individual Xk in OPOP .

Evolution loop

Set generations count index g = 1. ◦ g 0 g 0 Set POP = POP and OPOP = OPOP . ◦ While g NGmax (stopping criterion): ◦ ≤ g g g Combine POP and OPOP to obtain a population equals to their union UPOP = ◦ POP g OPOP g . ∪ Apply a fast non–dominated sorting procedure on UPOP g and identify the new non– ◦ g g dominated fronts 1 ,..., ND . {F F } g g Compute and assign the crowding–distance value to each individual Xk in UPOP and ◦ g sort each non–domination–ranked group ℓ ℓ 1,...,ND . {F } ∈{ } g g+1 Select the best NP individuals Xk to create the next population of parents POP . ◦ Apply binary tournament selection to POP g+1 to generate an intermediate population ◦ POP g+1' of size NP. Reproduction 3 Renewable DG–integrated electric power network planning 37

◃ Apply mutation and crossover operators to POP g+1', to create an offspring population OPOP g+1 of size NP. g+1 g+1 ◃ Evaluate the objective functions for each individual Xk in OPOP . g Set g = g +1 and verify the stopping criterion, if g > NGmax then return POP , selecting ◦ g the best front 1 as the optimal set of non–dominated solutions, otherwise continue the evolution loop.F

In correspondence to the nomenclature used in this thesis, the process of searching the set of non–dominated solutions carried out by the NSGA–II MCS–OPF is presented schematically in Figure 3.1. 38 3 Renewable DG–integrated electric power network planning

Set the values of NP, NGmax , pC , pM

R Generate randomly NP individuals [Q ], according to constraints (2.3), (3.3) 0 R 0 R 0 R 0 and (3.4), to form the initial population POP = [Q ]1,..., [Q ]k,..., [Q ]NP { }

Ω Ω Ω Ω Evaluate the objective functions βE(CG ) + (1 β)CVaRα(CG ) (3.1) and βE(ENS ) + (1 β)CVaRα(ENS ) (3.2) R 0 − 0 0 M− R 0 for each individual [Q ]k through MCS–OPF([Q]k, [Y ], Ω) with [Q]k = [Q [Q ]k] |

R 0 0 Rank the individuals [Q ]k in POP , applying a fast non–dominated 0 0 sorting procedure and identify the non–dominated fronts 1 ,..., ND {F F }

R 0 0 Compute and assign the crowding–distance value (dC ) to each individual [Q ]k in POP and sort, in ascending order 0 with respect to dC , the individuals belonging to the same non–domination–ranked group ℓ ℓ 1,...,ND {F } ∈{ }

0 0 Apply binary tournament selection to POP based on dC to generate an intermediate population POP ' of size NP

Apply mutation and crossover operators to POP0', to create an offspring population OPOP0 of size NP

R 0 0 Evaluate the objective functions (3.1) and (3.2) for each individual [Q ]k in OPOP

g 0 g 0 Set generations count index g = 1, POP = POP and OPOP = OPOP

g g g Define a union population as UPOP = POP OPOP ∪

R g g Rank the individuals [Q ]k in UPOP , applying a fast non–dominated sorting g procedure and identify the non–dominated fronts ℓ ℓ 1,...,ND {F } ∈{ }

R g g g Compute and assign dC to each individual [Q ]k in UPOP and sort each ℓ ℓ 1,...,ND {F } ∈{ }

R g g+1 Select the best NP individuals [Q ]k to create the next population of parents POP

Apply binary tournament selection to POP g+1 to generate an intermediate population POP g+1' of size NP

Apply mutation and crossover operators to POP g+1', to create an offspring population OPOP g+1 of size NP

R g+1 g+1 Evaluate the objective functions (3.1) and (3.2) for each individual [Q ]k in OPOP

Set g = g + 1

no Is g > NGmax ?

yes

g g Return POP and the optimal set of non–dominated solutions 1 F Figure 3.1 Flow chart of the proposed NSGA–II MCS–OPF framework 3 Renewable DG–integrated electric power network planning 39

3.2.2 MOO differential evolution–based approach

DE is a population–based and parallel, direct search method, shown to be one of the most efficient evolutionary algorithms to solve complex optimization problems [93, 96, 97]. The implementation of the original version of DE involves two main phases: initialization and evolution. The extended MOO–DE, with fast non–dominated sorting and crowding–distance comparison operator [92] is summarized below [93].

Initialization

Set the values of parameters: ◦ ◃ NP: population size.

◃ NGmax : maximum number of generations.

◃ pC : crossover coefficient [0, 1]. ∈ ◃ F: differential variation amplification factor [0, 2]. ∈ Form the initial population POP0, randomly generating NP decision matrices (individu- ◦ 0 0 0 0 als) X within the feasible space, POP = X1 ,..., Xk ,..., XNP 0 { 0 } 0 Evaluate the objective functions f1(Xk ),..., fNF (Xk ) for each individual Xk , where NF ◦ – number of objective functions{ } 0 Rank the individuals in POP , applying a fast non–dominated sorting [92] procedure ◦ with respect to the values of the objective functions and identify the non–dominated 0 0 0 0 fronts 1 ,..., ND , where 1 is the best front and ND the less good front. {F F } F F 0 Compute and assign the crowding–distance value (dC ) [92] to each individual Xk in ◦ 0 POP and sort, in ascending order with respect to dC , the individuals belonging to the 0 same non–domination–ranked group ℓ ℓ 1,...,ND . {F } ∈{ } Evolution loop

Set generations count index g = 1. ◦ g 0 Set POP = POP . ◦ While g NGmax (stopping criterion): ◦ ≤ Set a repository population RPOP as empty. ◦ Trial loop g g For each individual Xk in POP , k 1, . . . , NP : ∀ ∈ { } ◃ Sample from the uniform distribution three integer indexes in 1,..., NP such that { g }g g k1 k2 k3 k and choose the corresponding three individuals X , X , X = = = k1 k2 k3 ̸ ̸ ̸ g ◃ Generate a mutant individual XMk according to the following mutation operator:

g g g g XMk = Xk + F(Xk Xk ) (3.13) 1 2 − 3 40 3 Renewable DG–integrated electric power network planning

g ◃ Apply crossover operator, initializing a randomly generated vector XCk , whose g g dimensionality n is the same as that of Xk and each coordinate xck,i follows a uniform distribution with outcome in [0, 1] i 1,..., n . In addition, generate ∀ ∈ { } randomly an integer index i∗ in 1,..., n from a uniform distribution to ensure { g } g that at least one coordinate from XMk is exchanged to form trial individual XTk , g whose coordinates x tk,i are defined as follows:

g g g xmk,i if xck,i pC or i = i∗ x t = ≤ (3.14) k,i  g g xk,i if xck,i > pC and i = i∗  ̸ Evaluate the objective functions for the trial individual f XT g ,..., f XT g ; if ◃  1( k ) NF ( k ) g g g g g { g g } XTk dominates Xk , i.e., f (XTk ) f (Xk ) , XTk replaces Xk in POP , otherwise g g { g ≺ } retain Xk in POP and save XTk in the repository population RPOP. Set a combined population UPOP as POP g RPOP and rank the individuals in UPOP, ◦ applying a fast non–dominated sorting procedure∪ and identify the new non–dominated g g fronts 1 ,..., ND . {F F } Compute and assign the crowding–distance value dC to each individual in UPOP and ◦ g sort each non–domination–ranked group ℓ ℓ 1,...,ND . {F } ∈{ } Set POP g as the first NP (best) individuals of the ranked and sorted population UPOP, ◦ g POP = Xk : Xk UPOP, k 1, . . . , NP { ∈ ∈ { }} g If the stopping criterion is reached return POP , otherwise set g = g + 1. ◦

Analogously to the NSGA–II based FW1 and, in correspondence to the nomenclature used in this thesis, search for the set of non–dominated solutions performed by MOO–DE MCS–OPF is presented schematically in Figure 3.2. 3 Renewable DG–integrated electric power network planning 41

Set the values of NP, NGmax , pC , F

R Generate randomly NP individuals [Q ], according to constraints (2.3) and 0 R 0 R 0 R 0 (3.9), to form the initial population POP = [Q ]1,..., [Q ]k,..., [Q ]NP { }

Ω Ω Evaluate the objective functions E(CG ) (3.6) and DCVaRα(CG ) (3.7) for each R 0 0 0 M R 0 individual [Q ]k through MCS–OPF([Q]k, [Y ], Ω) with [Q]k = [Q [Q ]k] |

g 0 Set generations count index g = 1 and POP = POP

Set k = 1 and repository population RPOP as empty

R g g From the individual [Q ]k in POP , k 1, . . . , NP , generate a trial R g ∀ ∈ { } individual [QT ]k by applying mutation (3.13) and crossover (3.14) Ω Ω operators and evaluate the objective functions E(CG ) and DCVaRα(CG ) g g M R g through MCS–OPF([QT]k , [Y ], Ω) with [QT]k = [Q [QT ]k ] |

R g The original individual [Q ]k is R g yes no g The trial individual [QT ]k R g R g retained in POP and the trial R g g [QT ]k dominates [Q ]k ? R g replaces [Q ]k in POP individual [QT ]k is saved in the repository population RPOP

Set k = k + 1

no Is k > NP? yes Set the combined population UPOP as POP g RPOP, apply fast non–dominated sorting ∪ g procedure on UPOP, within each non–domination–ranked group ℓ ℓ 1,...,ND , compute {F } ∈{ } and assign the crowding–distance values dC and sort the individuals in descending order

g R R Set POP = [Q ]k : [Q ]k UPOP, k 1, . . . , NP { ∈ ∈ { }}

Set g = g + 1

no Is g > NGmax ?

yes

g g Return POP and select the best front 1 as the optimal set of non–dominated solutions F Figure 3.2 Flow chart of the proposed MOO–DE MCS–OPF framework

4 Computational Challenge

The computational challenge here presented consists in improving the performance of HO techniques used to solve the complex optimization problem of DG planning, when the objective function(s) is(are) evaluated by time consuming computational models, such as the developed MCS–OPF. For this, the integration of clustering into the HO search engine is considered.

4.1 Clustering in heuristic optimization

Clustering techniques like, k–means, fuzzy c–means and hierarchical clustering, among others, can be directed to the enhancement of the global and/or local searching ability of HO algorithms, and amounts to identifying groups of similar individuals and applying different evolution operators to those of a same cluster (group), e.g. for random generation of new individuals in the neighborhood of cluster centroids, or multi-parents crossover over new randomly generated individuals spread in the global feasible space [96–101]. In the literature, these approaches have been proved to be effective in improving convergence, but for ‘light-weight’ benchmark or not simulation–based objective function(s). When the computational time complexity associated to the evaluation of the objective function(s) is significant, even if convergence is improved by applying some ofthese methodologies, which imply a temporarily increment of the overall size of the population, the computational effort benefits may result counteracted. In addition, the accuracy of the clusters structures in representing the distribution of individuals must be controlled for performing clustering conveniently.

The main original contribution of the work here presented, lies in the development of the clustering strategy in a controlled manner. The implementation of such clustering strategy is done within a differential evolution (DE) optimization framework MCS–OPF model for the integration of renewable DG into an electric power system. The introduction of the clustering is hierarchically (i.e., hierarchical clustering analysis, HCA, [102]) by a controlled way of reducing the number of individuals to be evaluated during the DE search, therefore, improving the computational efficiency. Henceforth, the method is called hierarchical clustering differential evolution (HCDE). 44 4 Computational Challenge

HCA is introduced to build a hierarchical structure of grouping individuals (DG–integrated network plans) of the population that present closeness under the control of a specific linkage criterion based on defined distance metrics [102]. The HCA outcomes are the linkage distances at which the grouping actions take place, defining the different levels in the hierarchical structure. Two control parameters are introduced in the HCA, the cophenetic correlation coefficient (CCC) and a cutoff level coefficient of the linkage distances in the hierarchical structure of the groups(pco). The CCC is a similarity coefficient that measures how representative is the proposed grouping structure by comparing their linkage distances with the original distances between all the individuals in the population. In the hierarchical structure, the linkage distance given by pco sets the level at which the groups formed below it are considered to be ‘close enough’ to constitute independent clusters. The two parameters allow HCDE to adapt itself in each generation of the search, ‘deciding’ whether to perform clustering if the CCC is greater than or equal to a preset threshold (CCCT ) and cutting the hierarchical structure in independent clusters according to the linkage distance given by pco. Then, the individual closest to the centroid of each cluster is taken as the feasible representative solution in the population that enters the evolution phase of the HCDE algorithm.

4.2 Hierarchical clustering & differential evolution

The original version of DE keeps the population size NP constant, making the computational performance dependent mainly on the number of objective function evaluations (NFE) carried out during the evolution phase of the algorithm. Then, the integration of HCA into DE is aimed at the reduction of the number of individuals that enter the evolution loop in each generation so as to decrease the number of objective function evaluations.

HCA links individuals or groups of individuals which are similar with respect to a specific property, translated into a metric of distance, obtaining a hierarchical structure. In practice, an agglomerative procedure is used, which in NZ = NP 1 steps z fuses the closest pair or individuals or groups of individuals through a linkage function, e.g.− single linkage (nearest neighbor distance), complete linkage (furthest neighbor), average linkage, among others, until the complete hierarchical structure is built. The base hierarchical clustering algorithm used in this study can be expressed as follows [102]:

Step 1: Given a population POP POP = X1,..., Xk,..., XNP , form the set of singleton groups { } = ı = Xk , ı = k 1,..., NP and calculate the linkage distances between all the CNP groups{C using{ }} the∀ average∈ { as linkage} function and the Euclidean distance as metric: where, z dı, – average of the Euclidean distances between all the individuals Xk and Xk' belonging to the groups ı and , respectively, N z – number of groups at step z and ı ,  – C C C |C | |C | cardinalities of the groups ı and , respectively. C C 4 Computational Challenge 45

0 dz dz dz 1,2 1, 1,N z ··· ··· C 1 ...... z 2 . . . . . dı, = (Xk Xk') (4.1)  z z  ı  0 d d Xk ı Xk'  − z ı, ı,N z |C ||C | X∈C X∈C Æ D =  ···. . C   0 .. .  N z = NP z + 1 (4.1a)   C −  .  X , X POP, z 1, . . . , NP 1 ,  .. dz  k k'  N z 1,N z  ∀ ∈ ∈ { − }  C −0 C  ı,  1, . . . , N z   ∈ { C }  

1 Step 2: Fuse the first pair of groups ı' and ', for which dı',' is the minimum distance 1 C C min(D ) and form a new group NP+1 = ı' ' . Update the set of groups replacing C {C ∪ C } 2 C ı' and ' by NP+1, and calculate the linkage distances D between all the NP 1 groups inC usingC (4.1C). − C 2 Step 3: Fuse the second pair of groups ı' and ' for which dı',' is the minimum distance 2 C C min(D ), and form a new group NP+2 = ı' ' . As in the preceding step, update the set of groups and calculate the linkageC distances{C ∪D C3 between} all the NP 2 groups in using (4.1). C − C . .

NP 1 Step NP 1: Fuse the last pair of groups with linkage distance dı',' − , forming the last group − 2NP 1 = ı' ' that contains all the individuals X . C − {C ∪ C } The outcoming hierarchical (or tree) structure can be reported as a sorted table containing the NP 1 linkage distances relative to each pairing action of individuals/groups and be graphically illustrated− as a dendrogram. Table 4.1 and Figure 4.1 present, respectively, the resultant linkage distances and dendrogram obtained from an example set of NP = 8 two-dimensional individuals X using the above introduced HCA algorithm.

Table 4.1 Example hierarchical structure outcome

Step z Group Groups linked Linkage distance

1 1 9 2 6 = X2 X6 d2,6 C {C ∪ C } {{ } ∪ { }} 2 2 10 3 4 = X3 X4 d3,4 C {C ∪ C } {{ } ∪ { }} 3 3 11 1 7 = X1 X7 d1,7 C {C ∪ C } {{ } ∪ { }} 4 4 12 5 8 = X5 X8 d5,8 C {C ∪ C } {{ } ∪ { }} 5 5 13 9 11 = X2, X6 X1, X7 d9,11 C {C ∪ C } {{ } ∪ { }} 6 6 14 10 12 = X3, X4 X5, X8 d10,12 C {C ∪ C } {{ } ∪ { }} 7 7 15 13 14 = X1, X2, X6, X7 X3, X4, X5, X8 d13,14 C {C ∪ C } {{ } ∪ { }} 46 4 Computational Challenge

7 5 d13,14 3.0 C15 X8 4 X5 X 3 2.5 6 d10,12 X4 C14 3 2.0 xj d 2 1.5 5 X2 d9,11 C13 4 X6 d X7 3 5,8 2 1 d1,7 C11 C12 1.0 d3,4 X1 1 C10 d2,6

0 0.5 C9 0 1 2 3 4 5 C2 C6 C1 C7 C3 C4 C5 C8 xi

Figure 4.1 Example dendrogram for average linkage HCA

HCA builds the hierarchical structure through a linkage function introducing in each grouping action a larger or smaller degree of distortion with respect to the original distances between (ungrouped) individuals. The measurement of this distortion is important and the cophenetic correlation coefficient (CCC)is introduced to evaluate how representative is the hierarchical structure proposed by the HCA. The CCC can be obtained from Equations (4.2) and (4.3) below [102].

1 ¯1 ¯ (dk,k' D )(lk,k' ) k

0 l1,2 l1,k' l1,NP . ···. . ···......   z∗ 0 lk,k' lk,NP lk,k' = dı, (4.3) ···. . =  .. .  L  0 .  z∗ = min z : Xk, Xk' NP+z (4.3a)  .  { ∈ C }  .. l  k, k' 1, . . . , NP , ı,  1, . . . , 2NP 1  NP 1,NP   −0  ∀ ∈ { } ∈ −    

¯1 1 where D – average of the original Euclidean distances dı, between all the individuals, lk,k' – linkage z∗ ¯ distance dı, where the pair of individuals Xk and Xk' become members of the same group and – L average of the resultant linkage distances lı, between all the individuals. Recalling that the aim of nesting HCA into DE is to increase the computational performance by decreasing the NFE (times that the MCS–OPF is run) in each generation g, the presetting of a threshold CCCT for the CCC value allows defining the level of representativeness required tothe 4 Computational Challenge 47 hierarchical structure proposed. If by applying HCA over the population POP g the corresponding g g CCC is such that CCC CCCT , the built hierarchical structure is considered an acceptable representation of the original distances≥ amongst the individuals and the selection of a particular partition of the sets of groups can be performed, i.e., the determination of a specific number of clusters. On g the contrary, if CCC CCCT , the hierarchical structure is considered not representative enough since it introduces unacceptable≤ distortion that may affect the global searching process in the HCDE.

Whether the hierarchical structure is accepted, the clustering process itself takes place. As before z stated, the HCA outcome linkage distances dı, define each level (height) at which a pairing action is carried out. If the hierarchical structure is ‘cut off’ at a specific linkage distance dco, all the groups that are formed below that level become independent clusters. In each generation g of HCDE, a dco relative to the HCA outcome linkage distances for the corresponding POP g , is determined from a z preset cut–off level coefficient pco of the linkage distances between the minimum dı, that correspond to the first pairing action and the distance to form at least four clusters needed to performthe mutation process in the HCDE. Thus, dco can be obtained from Equation (4.4). Figure 4.2 shows the cut–off distance representation for the example aforementioned, for which the formed clusters are 2, 6 , 1 , 7 , 3, 4 , 5 and 8 . {C C } {C } {C } {C C } {C } {C }

dco = dmin + pco(dNC=4 dmin) (4.4) − z dmin min d (4.4a) = z ı,

dNC=4 = d 4 (4.4b) 1 NP %ile − z where, dmin – minimum linkage distance dı, that correspond to the first pairing action and dNC=4 – distance to form at least four clusters.

3.0

2.5

2.0 d 1.5 d NC =4 1.0 d co dmin 0.5 C2 C6 C1 C7 C3 C4 C5 C8

Figure 4.2 Example of cutoff distance calculation

The integration of HCA into DE and the definition of the parameters CCCT and pco allow HCDE adaptation at each generation, i.e., deciding whether to perform HCA and determining the clusters 48 4 Computational Challenge to be taken. Then, the individuals closest to the centroids of the formed clusters are considered as the representatives of the group which they belong to and are taken in a reduced population that enters the evolution phase of the HCDE. The proposed HCDE algorithm is summarized schematically in the flowchart of Figure 4.3. 4 Computational Challenge 49

Set the values of NP, NGmax , pC , F, CCCT , pco

R Generate randomly NP individuals [Q ], according to constraints (2.3), (3.11) 0 R 0 R 0 R 0 and (3.12), to form the initial population POP = [Q ]1,..., [Q ]k,..., [Q ]NP { }

Ω R 0 Evaluate the objective function ECG (3.10) for each individual [Q ]k 0 0 M R 0 through MCS–OPF([Q]k, [Y ], Ω) with [Q]k = [Q [Q ]k] |

g 0 Set generations count index g = 1 and POP = POP

z Perform HCA using the average distance linkage function obtaining dı',' by (4.1) for the NP 1 pairing actions z and calculate the cophenetic correlation index CCC g −

g Is CCCT CCC ? no ≤ yes g Cut off the hierarchical structure according to pco by (4.4) forming NP groups . R g {C } Obtain the individual [Q ]c closest–to–the–centroid for each ı , g g R g {CR g } ı 2NP 1, . . . , 2NP NP and set POP ∗ = [Q ]c ,..., [Q ]c g ∈ { − − } { 1 NP }

g g g R g R g R g Set NP = NP and POP ∗ = POP [Q ]1 ,..., [Q ]k ,..., [Q ]NP { }

Set k' = 1

R g g R g Perform the trial, for [Q ]k' POP ∗, generate a trial individual [QT ]k' by applying mutation (3.13) and crossover∈ (3.14) operators and evaluate the objective function ECGΩ

yes no Ω R g Ω R g Is ECG ([QT ]k') < ECG ([Q ]k')?

R g The trial individual [QT ]k' The original individual g g Set k' k' 1 R g g replaces QR QR in POP g = + Q is retained in POP [ ]k = [ ]k' [ ]k=k' no Is k' > NP g ? yes Set g = g + 1

no Is g > NGmax ?

yes

g Ω R g Sort the individuals in POP in descending order according to their values of ECG and return [Q ]1

Figure 4.3 Flow chart of the proposed HCDE framework

5 Applications

In this Chapter, examples of application of the proposed frameworks are given. The main results obtained and the derived insights are presented, while encouraging the interested reader to consult the corresponding Papers (i)–(iii) reported in Part II for further details.

For practical ease of the presentation of the examples, tables and figures reporting the data associated to each application are provided in the corresponding AppendicesA,B andC.

5.1 A risk–based MOO and MCS–OPF simulation framework for the integration of renewable DG and storage devices

This section introduces an example of application of the simulation and MOO framework for the integration of renewable DG and storage devices into an electric distribution network. The framework searches for the optimal size and location of the DG units, taking into account different sources of uncertainty: renewable resources availability, components failure and repair events, loads and grid power supply. The evaluation of the network performance is run by the developed MCS–OPF computational model and, as a response to the need of monitoring and controlling the risk associated to the performance of the optimal DG–integrated networks, the CVaR is integrated into the MOO optimization strategy which aims at the concurrent minimization of the expected values of CG and ENS, namely ECG and EENS, combined with their respective CVaR values, CVaR(CG) and CVaR(ENS), and weighed by a factor beta which allows modulating expected performance and risk. The MOO is performed by the NSGA–II presented in Chapter3, Section 3.2 and considering all the formulation relative to framework FW1 that can be found in the same Chapter3, Subsection 3.1.1.

5.1.1 IEEE 13–bus test feeder case

The framework FW1 is applied to a distribution network derived from the IEEE 13 nodes [2, 103]. The spatial structure of the network has not been altered but the the regulator, capacitor, switch feeders of zero length are neglected. The network is chosen purposely small, but with all relevant characteristics for the analysis, e.g. comparatively low and high spot and distributed load values 52 5 Applications and the presence of a power supply spot [103]. The original IEEE 13–bus test feeder is dimensioned such that the total power demand is satisfied without lines overloading. This is modified so thatit becomes of interest to consider the integration of renewable DG units. Specifically, the location and values of some of the load spots and the power ratings values of some feeders have been modified in order to generate conditions of power congestion of the lines, leading to shortages of power supply to specific portions of the network.

The distribution network presents a radial structure of n = 11 nodes and, therefore, (n 1) = 10 − feeders, as shown in Figure 5.1. The nominal voltage is V = 4.16 (kV), constant for the resolution of the DC–OPF problem. Four types of renewable DG technologies are considered to be integrated: solar photovoltaic (PV), wind turbines (W), electric vehicles (EV) and storage devices (ST).

5 4 i = 1 9 8 2 MG 6 10 3 11 7 i: node index MG: Main Supply spot spot load (kW)

Figure 5.1 Radial 11–nodes distribution network

Five optimizations runs of the NSGA–II with the nested MCS–OPF algorithm are performed, each one with a different value of the weight parameter β {1, 0.75, 0.5, 0.25, 0}, to analyze different tradeoffs between optimal average performance and∈ risk. From the equations that define the objective functions, (3.1) and (3.2), note that the value β = 1 corresponds to optimizing only the expected values ECG and EENS, whereas β = 0 corresponds to the opposite extreme case of optimizing only the CVaR values. Each NSGA–II run is set to perform NGmax = 300 generations over a population of NP = 100 chromosomes and, for the reproduction, the single–point crossover and mutation genetic operators are used. The crossover probability is pC = 1, whereas the mutation probability is pM = 0.1; the mutation can occur simultaneously in any bit of the chromosome.

Finally, NS = 250 random scenarios are simulated by the developed MCS–OPF with time step H ∆t = 1 (h), over an horizon of analysis of 10 years (t = 87600 (h)), in which the investment and fixed costs are prorated hourly.

5.1.2 Conditional value–at–risk: expected performance and risk trade–off

The Pareto fronts resulting from the NSGA–II MCS–OPF are presented in Figure 5.2 for the different values of β. The ‘last generation’ population is shown and the non–dominated solutions are marked in bold. Each non–dominated solution in the different Pareto fronts corresponds to an optimal 5 Applications 53

R decision matrix [Q ] for the sizing and allocation of DG, i.e., an optimal DG–integrated network M R configuration ([Q], Y ) where Q = [Q Q ]. { } |

167.5 β = 1 167.5 β = 0.75 175.0 β = 0.5

) ($) )

(

α (

160.0 162.5 α 170.0

($)

CVaR

0.5

ECG 0.25

152.5 157.5 165.0

ECG+

0.5 0.75 145.0 152.5 160.0 650 725 800 875 750 850 950 1050 800 950 1100 1250

EENS (kWh) 0.75 EENS+0.25 CVaRα(ENS) (kWh) 0.5 EENS+0.5 CVaRα(ENS) (kWh)

180.0 β = 0.25 185.0 β = 0

) ($) )

CG (

α 175.0 180.0

) ($) )

CVaR

(

α 0.75

170.0 175.0

CVaR

ECG+ 0.25 165.0 170.0 1000 1100 1200 1300 1000 1200 1400 1600

0.25 EENS+0.75 CVaRα(ENS) (kWh) CVaRα(ENS) (kWh)

Figure 5.2 Pareto fronts for different values of β

175 A 175 B

Ø∝CVaRα(ENS) Ø∝CVaRα(CG)

165 165 betaβ = 1 1

betaβ = 0.75 075

($) ($)

betaβ = 0.5 05 ECG ECG betaβ = 0.25 025 155 155 betaβ = 0 0 msMG

145 145 600 800 1000 1200 600 800 1000 1200 EENS (kWh) EENS (kWh)

Figure 5.3 Bubble plots EENS v/s ECG. Diameter of bubbles proportional to CVaR(ENS) (A) and CVaR(CG) (B) 54 5 Applications

In the Pareto fronts obtained, three representative non–dominated solutions are looked for the R † analysis: those with minimum values of the objective functions independently, denoted [Q ]minf (CG) R ‡ and [Q ]minf (ENS) , respectively, and an intermediate compromised solution at the ‘elbow’ of the R Pareto front, [Q ]f (CG) f (ENS). Table 5.1 presents the values of the objective functions, ECG, ENS and their respective CVaR| values for the selected solutions. The ECG, EENS and CVaR values of the case in which no DG is integrated in the network (MG case) is also reported.

Figure 5.3 shows a bubble plot representation of the selected optimal solutions. The axes report the EENS and ECG values while the diameters of the bubbles are proportional to their respective CVaR values. The MG case is also plotted.

Table 5.1 Objective functions: expected and CVaR values of selected Pareto front solutions

β f (CG) ($) f (ENS) (kW h) ECG ($) CVaR(CG) ($) EENS (kW h) CVaR(ENS) (kW h) MG – – – 170.27 179.24 1109.21 1656.53 QR min 160.91 666.95 160.91 185.11 666.95 1093.12 [ ]f (CG) QR f (CG) 1 150.83 671.05 150.83 179.47 671.05 1185.53 [ ]f (ENS) QR min 148.68 726.57 148.68 178.23 726.57 1279.37 [ ]f (ENS) QR min 166.41 797.07 160.68 183.62 677.74 1155.11 [ ]f (CG) QR f (CG) 0.75 159.35 805.27 153.09 178.15 697.17 1129.62 [ ]f (ENS) QR min 155.61 867.08 147.66 179.45 729.81 1278.94 [ ]f (ENS) QR min 171.54 868.61 159.43 183.64 641.68 1095.52 [ ]f (CG) QR f (CG) 0.5 166.67 936.58 154.67 178.53 701.72 1171.47 [ ]f (ENS) QR min 162.99 1131.64 150.45 175.58 843.53 1419.79 [ ]f (ENS) QR min 172.95 1033.65 156.55 178.42 723.19 1137.18 [ ]f (CG) QR f (CG) 0.25 171.25 1076.53 156.32 176.24 743.61 1187.43 [ ]f (ENS) QR min 169.07 1207.33 158.64 173.47 835.23 1331.34 [ ]f (ENS) QR min 179.03 1144.36 163.82 179.03 744.71 1144.31 [ ]f (CG) QR f (CG) 0 176.62 1197.79 160.93 176.62 749.21 1197.74 [ ]f (ENS) QR min 172.87 1307.33 159.78 172.87 828.55 1307.35 [ ]f (ENS)

From Table 5.1 and Figure 5.3 it can be seen that, the MG case has an expected performance (EENS = 1109.21 (kW h) and ECG = 170.27 ($)) inferior (high EENS and ECG) to any case for which DG is optimally integrated. Furthermore, the CVaR(ENS) = 1656.53 (kW h) for the MG case is the highest, indicating the high risk of actually achieving the expected performance of ENS. This confirms that DG is capable of providing a gain of reliability of power supply and economic benefits, the risk of falling in scenarios of large amounts of ENS being reduced.

Comparing among the selected optimal DG–integrated networks, in general the expected performances of EENS and ECG are progressively lower for increasing β. This to be expected: lowering R the values of β, the MOO tends to search for optimal allocations and sizing [Q ] that sacrifice expected performance at the benefit of decreasing the level of risk (CVaR). These insights canserve

† f (CG) = βECG + (1 β)CVaR(CG) by (3.1) ‡ f (ENS) = βEENS +− (1 β)CVaR(ENS) by (3.2) − 5 Applications 55 the decision making process on the integration of renewable DG into the network, looking not only at the give–and–take between the values of EENS and, but also at the level of risk of not achieving such expected performances due to the high variability.

5.1.3 Optimal DG–integrated network plans

Figure 5.4 shows the average total DG power allocated in the distribution network and its breakdown R by type of DG technology for the optimal [Q ] as a function of β. It can be pointed out that the contribution of EV is practically negligible if compared with the other technologies. This is due to the fact that the probability that the EV is in a discharging state is much lower than that of being in the other two possible operating states, charging and disconnected (see Figure A.2 in AppendixA), combined with the fact that when EV is charging the effects are opposite to those desired.

β B C

PPV PW

PT PT

β β D E

PEV PST

PT PT

β β

Figure 5.4 Average total DG power allocated (A) and its breakdown by type of DG: PV (B), W (C), EV (D) and ST (E)

The analysis of the results for different β values also allows highlighting the impact that each type of renewable DG technology has on the network performance. As can be noticed in Figure 5.4(A), the average total renewable DG power optimally allocated, increases progressively for increasing values 56 5 Applications of β: this could mean that to obtain less ‘risky’ expected performances less renewable DG power needs to be installed. However, focusing on the individual fractions of average power allocated by PV, W and ST (Figure 5.4(B), (C) and (E), respectively), show that a reduction of the risk in the

EENS and ECG is achieved specifically diminishing the proportion of PV power (fromβ 0.29 =1 to

0.11β=0) while increasing the W and ST (from 0.38β=1 to 0.48β=0 and from 0.31β=1 to 0.39β=0, respectively), but this increment of W and ST power is not enough to balance the loss of PV power due to the limits imposed by the constraints in the number of each DG technology to be installed given by τj. Thus, PV power supply is shown to most contribute to the achievement of optimal expected performances, but with higher levels of risk. On the other hand, privileging the integration of W and ST power supply provides more balanced optimal solutions in terms of expectations and of achieving these expectations.

Table 5.2 summarizes the minimum, average and maximum total renewable DG power allocated per node. The tendency is to install more localized sources (mainly nodes 4 and 8) of renewable DG power when the MOO searches only for the optimal expected performances (β = 1) and to have a more uniformly allocation of the power when searches for minimizing merely the CVaR (β = 0).

Table 5.2 Average, minimum and maximum total DG power allocated per node

β ¯ PT (kW) 1 0.75 0.5 0.25 0 node min mean max min mean max min mean max min mean max min mean max 1 12.08 34.44 54.77 1.15 22.40 38.56 0.00 19.23 40.98 0.00 39.03 121.00 3.00 17.33 34.71 2 2.30 40.72 69.73 0.00 49.95 77.70 36.50 58.40 123.36 3.00 63.61 132.93 0.00 42.54 84.09 3 0.00 24.83 46.45 14.80 41.79 85.03 0.00 37.94 105.11 4.00 36.87 98.53 1.00 32.84 77.78 4 76.00 110.00 133.41 1.15 67.40 133.63 0.58 38.04 80.13 6.15 20.73 61.85 0.00 39.85 85.86 5 22.60 52.39 77.08 28.90 60.66 98.59 12.63 89.39 143.50 3.30 23.49 54.25 1.00 24.97 79.64 6 12.33 55.56 85.46 10.45 21.22 38.95 2.00 27.68 106.26 12.15 53.78 84.43 0.00 50.64 116.85 7 8.00 16.52 35.38 39.38 64.07 104.05 0.00 52.03 159.73 0.00 34.09 92.81 5.00 18.51 39.23 8 79.03 111.20 146.63 30.00 74.57 114.41 0.00 40.60 146.06 4.00 37.94 102.60 1.00 39.49 119.38 9 0.00 20.03 68.73 4.00 74.07 107.88 0.00 46.72 85.61 0.00 44.06 94.08 0.00 32.86 74.53 10 0.00 9.07 25.35 0.00 1.58 7.88 0.00 11.88 58.69 0.00 8.58 43.40 0.00 30.12 83.45 11 0.00 9.98 17.68 0.00 3.04 13.20 0.00 4.74 23.45 0.00 8.99 45.95 0.00 7.31 51.17

5.1.4 Brief summary

The results obtained show the capability of the framework FW1 to identify Pareto optimal sets of renewable DG units allocations. The integration of the conditional value–at–risk into the framework and the performing of optimizations for different values of the weight parameter β has shown the possibility of optimizing expected economic and reliability of power supply performances while controlling the risk in its achievement. The contribution of each type of renewable DG technology 5 Applications 57 can also be analyzed, in terms of size and location, indicating which is more suitable for specific preferences of the decision makers.

5.2 Hierarchical clustering analysis and differential evolution (HCDE) for optimal integration of renewable DG

The current Section presents an example of application relative to the HCDE framework developed to address the computational challenge given by the expenses of large computational times required to solve DG planning optimization problems under uncertainty. Coherently to the presentation of HCDE framework in Chapter4 and the single optimization strategy formulated in Chapter3, Subsection 3.1.1 (FW3), the designed search engine embeds HCA within a DE search scheme to identify groups of similar individuals in the DE population to, then, calculate the objective function to be minimized, ECG, only for selected representative individuals of the groups. The HCDE evaluates the objective function by MCS–OPF and performs HCA in a controlled manner, by presetting two parameters: a threshold to the cophenetic correlation coefficient CCCT and cutoff level coefficient pco; which allows to decide whether or not is convenient to perform clustering and at what scale, respectively.

5.2.1 IEEE 13–bus test feeder case

This example regards a modification of the IEEE 13–bus test feeder distribution network [2] with the original spatial structure but neglecting the feeders of length zero, the regulator, capacitor and switch. The resulting network has n = 11 nodes and presents the relevant characteristics of interest for the analysis, e.g. the presence of a bulk–power supply spot and comparatively low and high NET spot, and distributed load values [103]. The nominal voltage V is 4.16 (kV), kept constant for the resolution of the DC optimal power flow problem. An schematic view of the network is show in Figure 5.5. Four types of renewable DG technologies are considered to be integrated: solar photovoltaic (PV), wind turbines (W), electric vehicles (EV) and storage devices (ST).

A total of NS = 500 random scenarios are simulated by the MCS–OPF with time step ∆t = 1 H (h), over a horizon of analysis of 10 years (t = 87600 (h)), in which the investment and fixed costs are pro–rated hourly.

The DE iterations are set to perform NGmax = 500 generations over five different cases of population NP 10, 20, 30, 40, 50 . The differential variation amplification factor F is 1 to ∈ { } R maintain the integer–valued definition of the individuals [Q ] after the mutation, whereas the crossover coefficient pC is 0.1. 58 5 Applications

MG i = 1

5 4 2 633-6343

9 8 671-6926 7

10 11

Figure 5.5 Radial 11-nodes distribution network

HCDE runs are performed under the same conditions set for DE (NGmax , F and pC ), but for the population size NP of 50 individuals. A sensitivity analysis is performed over the HCA control parameters, CCCT and linkage distances cutoff level coefficient pco, for all the nine possible pairs (CCCT , pco) with CCCT 0.6, 0.7, 0.8 and pco 0.25, 0.50, 0.75 . Finally, for each of the five DE and nine HCDE settings,∈ 20 { realizations} are carried∈ { out. }

5.2.2 Quantification of the benefits of HCDE

The results of the DE MCS–OPF for the different population sizes NP 10, 20, 30, 40, 50 are shown in Figure 5.6. The 50th percentile (%ile) or median values of the∈ { minimum global} costs

ECGmin, obtained from each experiment with fixed values of NP, are presented as functions of the respective numbers of objective function evaluations NFE; the error bars represent the 15th and 85th %iles.

15-85th %iles

50th %ile min

Figure 5.6 ECGmin vs NFE for NP 10, 20, 30, 40, 50 set in DE ∈ { } 5 Applications 59

As expected, for the same number of generations set in the DE MCS–OPF, the larger the population size considered the lower the values of ECGmin obtained (better ‘quality’ of the minimum). Additionally, It can be observed marked tendencies in the reduction of both median and 15–85th

%iles values of ECGmin for increasing NFE. Performing a curve fitting over these values, the 0.13 0.115 following curves are gotten: ECGmin;50th%ile = 49.07NFE− , ECGmin;15th%ile = 49.07NFE− 0.118 2 and ECGmin;85th%ile = 49.07NFE− , with the respective coefficients of determination R50th%ile = 2 2 0.994, R15th%ile = 0.998 and R85th%ile = 0.998. The fact that the difference between the values of the 15–85th %iles is constant indicates that the dispersion in the ECGmin(NFE) does not depend on NP and can suggest that the global searching carried out by the DE is performed homogeneously in the feasible space that contains multiple local minima.

(NP, CCCthT , pco) 50th %ile min

Figure 5.7 ECGmin vs NFE for each (NP, CCCT , pco) set in HCDE

Figure 5.7 reports the median ECGmin values corresponding to the HCDE MCS–OPF realizations superposed to the distribution of the median ECGmin and 15–85th %iles values of the base DE experiments represented by the square markers and shaded area, respectively. The vertical and horizontal error bars account for the 15–85th %iles of the outcome ECGmin and NFE values.

Focusing on CCCT , it can be noticed that for the two extreme cases, CCCT = 0.6 and 0.8, the dispersion of the number of objective function evaluations is relatively small. On the contrary, the cases with a CCCT = 0.7 present high variability. This can be explained by the behavior of the CCC along each generation g in the evolution loop. Figure 5.8 shows the median, 15th and 85th %iles CCC values as a function of generation g derived from all HCDE MCS–OPF realizations. On the one hand, recalling that CCCT is used to control whether it is convenient to perform HCA, the small NFE dispersion in the case with CCCT = 0.6 is because clustering is practically been applied in all g generations (CCCT CCC ), thus disabling any effect generated by passing from populations with original size NP to≤ reduced populations with NP g NP and vice versa. On the other hand, the ≤ effect is also being avoided in the case CCCT = 0.8 by not applying clustering. Indeed, in Figure 5.8 60 5 Applications it can be observed that after the generation number 50 it is unlikely that by performing HCA the proposed hierarchical grouping structures represent well enough the population.

50th %ile

15-85th %iles g

Figure 5.8 CCC behavior per generation g

Differently, the cases for which CCCT = 0.7 present high dispersion in the NFE since the median values of CCC g move in the neighborhood of the threshold throughout the major part of the evolution loop in the HCDE. Moreover, in general terms, the values of CCC g 15–85th %iles maintain certain symmetry with respect to the median, i.e., performing or not HCA are equally likely events, producing high fluctuations in the number of individuals considered as population and, therefore, affecting in the same way the NFE.

(50, 0.6, 0.25) (50, 0.7, 0.25) (50, 0.8, 0.25) (50, 0.6, 0.50) (50, 0.7, 0.50) (50, 0.8, 0.50) (50, 0.6, 0.75) (50, 0.7, 0.75) (50, 0.8, 0.75)

g Figure 5.9 Empirical NP pdf for each (NP, CCCT , pco) set in HCDE

The above mentioned insights are noticeable also in Figure 5.9, which shows the empirical g probability density functions (pdfs) of the population size NP per generation for each (NP, CCCT , pco) set in HCDE. Indeed, the average probabilities of performing HCA throughout the evolution cycle for the different values of CCCT = 0.6, 0.7 and 0.8 are 0.980, 0.540 and 0.078, respectively. 5 Applications 61

Regarding the cutoff level coefficient of the linkage distances pco, in Figure 5.9 it is possible to identify the three peaks of reduction in the population size, confirming the role of this control parameter in defining the level at which the hierarchical structures proposed are ‘cut off’ whenthe

HCA takes place. In fact, lower values of pco imply smaller reduction in the population size because of the higher demand of proximity between individuals or groups of individuals. In the opposite side, higher values of pco allow forming clusters from individuals or groups which are relatively less similar.

From the results obtained for all the different DE and HCDE settings, six representative cases are the focus of the analysis (Figure 5.7). From the DE runs, the settings with extreme and middle population size NP 10, 30, 50 are selected, whereas from HCDE, the cases (NP, CCCT , pco) set as (50, 0.6, 0.25),∈ (50,{ 0.6, 0.50),} (50, 0.7, 0.50) and (50, 0.7, 0.75) are chosen. The former (50, 0.6, 0.25) and (50, 0.6, 0.50) cases present significant reductions in the number of NFE, with small dispersion and loss of quality of the ECGmin obtained, compared to the results regarded by diminishing directly the fixed NP in DE from 50 to 10. Similarly, the cases (50, 0.7, 0.50) and

(50, 0.7, 0.75) may lead to considerable reductions in NFE, with acceptable losses of ECGmin, but subject to a high degree of variability that compromises the performance.

As for computational times, running on an Intel® Core™i7-3740QM (PC) 2.70 GHz without performing parallel computing, the average time to evaluate the objective function is 4.592 (s) for the NS = 500 scenarios in the MCS–OPF; for a fixed population of NP = 50 and its corresponding NFE = 20050, the total time for a single run is on average 25.574 (h). Taking into account this, under commonly used hardware configurations, the reductions in computational time that canbe achieved by using HCDE with (50, 0.6, 0.25) and (50, 0.6, 0.50) settings are 19% and 49% for the median, 23% and 51% for the 15th %ile, and 16% and 43% for the 85th %ile, respectively.

5.2.3 Identification of time complexity conditions and limits

The integration of HCA into the DE algorithm introduces a significant time complexity, conditioning the reductions of computational efforts that can be obtained by applying the proposed HCDE MCS– OPF framework. Indeed, if performing HCA along all generations of DE and running the MCS–OPF on an eventually reduced population (depending on CCCT and pco) is computationally heavier than running the MCS–OPF over the complete population, the effects of the framework can be negligible or even negative. It is possible to formulate the condition to obtain reductions in the computational efforts by the proposed HCDE MCS–OPF framework, from the asymptotic time complexities of the main algorithms that compose it. Table 5.3 reports the independent asymptotic time complexities as functions of the generic size m of the input to each algorithm and of the parameters that define the dimensionality of the HCDE MCS–OPF framework [102, 104]. 62 5 Applications

Table 5.3 Asymptotic time complexity of the algorithms

Algorithm PDIST HC MCS OPF Time complexity T 2 a 2 b O(dm ) O(m log(m)) O(m) O(size(A)) 2 2 2 O(n(m + r) NP ) O(NP log(NP)) O(NS n(m + r)) O(NS (n(m + r)) ) × × × where n(m + r) represents the size of the DG-integrated network, i.e., the number of nodes n times the number of all the types or technologies of power generation available, m types bulk–power suppliers and r DG types, NP is the size of the complete population and NS is the number of scenarios in the MCS–OPF. a Pairwise distance PDIST between all m vectors of size d. b The matrix A comes from the canonical form Ax b of the linear programming of the DC–OPF problem approximation. ≤

Comparing the asymptotic time complexities of the algorithms involved in the realization of the proposed framework with and without integrating HCA, the following inequalities must be fulfilled in order to obtain a reduction in the computational time by HCDE:

PDIST HC MCS–OPF MCS–OPF g T(n(m + r), NP) + T(NP) +E(NP ) T(NS, n(m + r)) < NP T(NS, n(m + r)) × × z }| { z }| { z }| { z }| { 2 2 g ⇓ 2 2 n(m + r) NP + NP log(NP) + E(NP ) NS (n(m + r)) < NP NS (n(m + r)) × × × × × ⇓ NP NP log(NP) κ = + + ε < 1 (5.1) NP n(m + r) NP (n(m + r))2 × × g E(NP ) n, (m + r), NP, NS Z∗, ε = (0, 1] ∀ ∈ NP ∈ where ε is the expected ratio of the population NP g evaluated along all generations g of DE to the total population NP and κ is the ratio of the asymptotic time complexities of HCDE to DE.

By Equation (5.1), the contribution of the terms related with the complexity of MCS–OPF, dependent on NS and n(m + r), is considerably large for the fulfillment of the inequality conditions. In fact, when using DE, it is commonly accepted to set a size of the population NP not greater than ten times the size of the decision variables, in this case, 10n(m + r) [93], making the first two terms of κ strongly dependent on the number of scenarios NS. Moreover, given the complexity of the general problem, higher values of NS lead to a better approximation of the objective function via MCS–OPF, i.e., the more likely is to fulfill the condition and the greater can be the reduction of computation time. However, the value of ε depends on the probability of performing clustering in each generation and at what level, controlled by CCCT and pco respectively. In some cases, ε can be close to 1 (as it is inferred from Figure 5.9) implying negligible benefits. Table 5.4 shows the 5 Applications 63

values of the ratio κ for each (NP, CCCT , pco) set in HCDE considering the dimensionality of the present case study defined by the values of the parameters n(m + r) = 55, NS = 500, NP = 50. The value of 1 κ can be interpreted as the expected asymptotic relative time reduction achieved by performing− HCDE.

Table 5.4 Ratio κ for each (NP, CCCT , pco)

g NP NP log(NP) E(NP ) (NP, CCCT , pco) ε = κ 1 κ NP n(m + r) NP (n(m + r))2 NP − (50, 0.6, 0.25) × × 0.817 0.819 0.181 (50, 0.7, 0.25) 0.921 0.923 0.077 (50, 0.8, 0.25) 0.987 0.989 0.011 (50, 0.6, 0.50) 0.510 0.512 0.488 (50, 0.7, 0.50)1.818e 03 3.418e 05 0.738 0.740 0.260 (50, 0.8, 0.50)− − 0.978 0.979 0.021 (50, 0.6, 0.75) 0.259 0.261 0.739 (50, 0.7, 0.75) 0.487 0.488 0.512 (50, 0.8, 0.75) 0.909 0.911 0.089

5.2.4 Optimal DG–integrated network plans

Figure 5.10 shows the average total DG power allocated in the distribution network and the corresponding investment costs of the DE and HCDE MCS–OPF cases selected, choosing the corresponding R optimal DG–integrated plans as the decision matrices [Q ] for which their ECGmin values are the closest to the median ECGmin value obtained for the twenty runs of each (NP, CCCT , pco) setting. It can be pointed out that in all the cases, the contribution of EV is practically negligible if compared with the other technologies. This is due to a combination of two facts: the probability that the EV is in a discharging state is much lower than that of being in the other two possible operating states, charging and disconnected (see Figure B.2 in AppendixB) and when EV is charging, the effects are

opposite to those desired, i.e., it is acting as loads.

($) CI CI

Figure 5.10 Average total DG power allocated and investment cost for representative (NP, CCCT , pco) settings 64 5 Applications

In all generality, both the investment cost CI and the average power installed by DG is comparable in all the cases, except for the setting (50, 0.7, 0.75) for which the level of clustering determined by pco = 0.75, that translates into higher reductions of the population size, may lead to less similar local minima than the other settings.

1 2 3 4 5 node i 6 7 8 9 10 11

Figure 5.11 Nodal average total DG power for representative (NP, CCCT , pco) settings

The average total renewable DG power allocated per node is summarized in Figure 5.11. Even R though all the ECG optimal decision matrices [Q ] show differences, the tendency is to install localized sources of renewable DG power between two identifiable portions of the distribution network, up and downstream the feeder (2,6) (Figure 5.5), giving preference to the second portion which presents higher and non-stream homogeneous nodal load profiles.

5.2.5 Brief summary

The results show that the the framework is effective in finding optimal DG–integrated network plans, with acceptable reductions of the computational efforts required while maintaining small dispersion and loss of quality in the minimum ECG obtained. The sensitivity analysis over the control parameters of the HCA suggest that the efficiency is improved with CCCT that allows the clustering in almost all generations along the DE search, setting the level of clustering to no more than the 50% (pco = 0.5) of the feasible linkage distances range in the hierarchical structure proposed. 5 Applications 65

5.3 A MOO and MCS–OPF simulation framework for risk–controlled integration of DG

In this Section, an example of application of the MOO framework for the integration of renewable DG into electric power networks is reported. The framework searches for the optimal size and location of different DG technologies, taking into account uncertain ties related to primary renewable resources availability, components failures, power demands and bulk-power supply. The network operation is emulated by the developed MCS–OPF computational model, assessing the system performance in terms of CG. To measure uncertainty, the DCVaR is introduced enabling the conjoint control of risk due to its axiomatic relation to the CVaR. A MOO strategy is adopted for the simultaneous minimization of the ECG and the associated deviation DCVaR(CG). This is operatively implemented by the MOO–DE described in Chapter3, Section 3.2 and respecting the formulation relative to framework FW2, to be found in the same cited Chapter.

5.3.1 IEEE 30 bus sub-transmission and distribution test system

The IEEE 30 bus sub-transmission and distribution test system is regarded for the analysis [11]. This constitutes a portion of the Midwestern U.S. electric power system which presents relevant characteristics of interest for the analysis, e.g., the presence of bulk-power supply spots different in type and with comparatively low and high nodal load profiles. An important consideration is that the synchronous condensers are neglected, given the DC assumptions made in the current thesis work for the resolution of the OPF problem.

The network consists of n = 30 nodes, a mesh deployment of 41 T&D lines and 2 transformers or bulk-power suppliers, as shown in Figure 5.12. The reference apparent power is Sre f = 100 (MV A). In this case, two renewable DG technologies are considered: solar photovoltaic (PV) and wind turbines (W).

Table 5.5 summarizes the main parameters set for the general MOO–DE and MCS–OPF framework.

Table 5.5 MOO-DE and MCS–OPF parameters

Parameter Nomenclature Value Population size NP 100

Maximum n◦ of generations NGmax 600

Crossover coefficient pC 0.1 Differential variation amplification factor F 1

N◦ of MCS-OPF scenarios NS 24000 Scenario duration (h) ∆t 1 66 5 Applications

30 29 THREE WINDING TRANSFORMER COVERDALE EQUIVALENTS 27 28 HANCOCK ROANOKE 12 10 C 9 C 13 4 11 26 6 24 25 33 (kV) 23 15 132 (kV) 14 18 Load 21 G Generator 19 22 C Synchronous condenser G 16 20 1 GLEN LYN 12 17 13 10 3 C HANCOCK 8 4 C 11 KUMIS ROANOKE 9 6 REUSENS 7 BLAINE 2 CLAYTOR G C FIELDALE 5

Figure 5.12 IEEE 30 bus sub-transmission and distribution test system diagram 5 Applications 67

5.3.2 Conditional value–at–risk deviation: expected performance, uncertainty and risk trade–off

The Pareto front resulting from the MOO-DE MCS–OPF is presented in Figure 5.13. The entire last generation population at convergence is shown by gray squares and the non–dominated solutions are the blue bullets. The base case (MG) in which no DG is integrated in the network is also shown R as dark gray triangle. Each solution in the Pareto Front corresponds to a decision matrix [Q ] that indicates the number of units and locations of the different types of DG technologies integrated in the network.

Recalling the relation given by equation (2.29), DCVaRα(x) = CVaRα(x E(x)), it is possible to draw a map of iso–CVaR curves in the plot of non–dominated solutions and so,− to include risk into the trade–off between expected performance and uncertainty, represented by ECG and DCVaRα(CG), respectively. Then, the compromised solution, namely QR min can be found, that minimizes [ ]CVaR(CG) risk as reported in Figure 5.14(A). The reciprocal case, i.e., a map of iso–DCVaR curves is drawn in the distribution of non–dominated solutions plotted as ECG vs DCVaRα(CG), and it is shown in Figure 5.14(B). Three representative non–dominated solutions are considered for the analysis: those with minimum values of ECG and DCVaR CG , denoted QR min and QR min respectively, α( ) [ ]ECG [ ]DCVaR(CG) and QR min which, as mentioned earlier, minimizes risk. [ ]CVaR(CG) 8.00 8.00

PF Dominated solutions 7.75 7.75 MG (no DG case) ) h / 7.50 $

k 7.50 (

) G C (

R 7.25 a 7.25 V C D

7.00 6.5000 7.00

6.3750

6.75 6.75 -12.50 -12.00 -11.506.2500 -11.00 -10.50 -12.50 -12.00 -11.50 -6.0000-11.00-5.9375 -10.50-5.8750 ECG (k$/h)

Figure 5.13 Set of non-dominated solutions: Pareto front 68 5 Applications

8.00 R min A Case []Q ECG Case []QR min R min CVaRCG() []Q ECG R min 7.75 Case []Q DCVaRCG()

Ca se M G

) h

/ iso CVaRCG() curvesa

k (

7.50

)

(

a R min 6.5000

7.25 []Q CVaRCG() DCVaR []QR min 7.00 DCVaR() CG 6.3750

6.75 6.2500 -12.50 -12.25 -12.00 -11.75 -11.50 -11.25 -6.0000-11.00 -5.9375-10.75 -5.8750-10.50 ECG (k$/h) -4.00 0.4500 B

0.4375 -4.25

) R min h

/ []Q ECG

(

) 0.4250 R min -6.0000 -5.9375 -5.8750

CG -4.50 []Q DCVaR() CG

(

R min Case []Q ECG

CVaR R min Case [Q ]CVaR() CG -4.75 R min Case [Q ]DCVaR() CG R min []Q CVaR() CG Case MG

iso DCVaRa ( CG ) curves

-5.00 -12.50 -12.25 -12.00 -11.75 -11.50 -11.25 -11.00 -10.75 -10.50 ECG (k$/h)

Figure 5.14 Set of non–dominated solutions with iso–CVaR (A) and iso–DCVaR (B) curves

R min R min In Figure 5.14, it can be observed that the three solutions of interest, [Q ]ECG, [Q ]CVaR and R min [Q ]DCVaR, lead to considerable improvements in expected performance and risk with respect to the base case MG, in which no renewable generation is integrated into the network. However, the level of uncertainty in the ECG estimation is increased, in all DG–integrated solutions, because of their stochasticity. Even so, in comparison to the MG case, the increase in the level of uncertainty for all DG–integrated cases (on average 1.067 (k$/h)) is much less than the gain in both expected 5 Applications 69 performance and risk (on average 6.035 and 4.967 (k$/h)), respectively). This fact can be seen also in the empirical CG probability− density functions− (pdf) shown in Figure 5.15.

0.08 DCVaRa(CG) = 6.379 (k$/h) ECG = -5.937 (k$/h) CVaR (CG) = 0.442 (k$/h) a Case MG 0.06 1.2E-03

0.04 8.0E-04

frequency 4.0E-04 0.02 0.0E+00 0 10 20 30 40 50 60 0

0.048 -30 DCVaR-20 a(CG-10) = 7.8620 (k$/h) 10 20 30 40 50 60 ECG = -12.281 (k$/h) CVaRa(CG) = -4.419 (k$/h) R min Ca se [Q ] ECG 0.036 1.2E-03

0.024 8.0E-04

frequency 4.0E-04 0.012 0.0E+00 0 10 20 30 40 50 60 0 0.048 -30 DCVaR-20 a(CG-10) = 7.3530 (k$/h) 10 20 30 40 50 60 ECG = -12.087 (k$/h) CVaRa(CG) = -4.733 (k$/h) R min CG (k$/h) Case []Q CVaR() CG 0.036 1.2E-03

0.024 8.0E-04

frequency 4.0E-04 0.012 0.0E+00 0 10 20 30 40 50 60 0 -30 -20 -10 0 10 20 30 40 50 60 0.048 DCVaRa(CG) = 7.125 (k$/h) ECG = -11.548 (k$/h) CVaRa(CGCG) = (k$/h)-4.422 (k$/h) R min Case []Q DCVaR() CG 0.036 1.2E-03

0.024 8.0E-04

frequency 4.0E-04 0.012 0.0E+00 0 10 20 30 40 50 60 0 -30 -20 -10 0 10 20 30 40 50 60 CG (k$/h)

Figure 5.15 Empirical CG probability density functions 70 5 Applications

Furthermore, it can be inferred from Figure 5.15 that, in general, all CG empirical pdfs show three main peaks. This is due to the characteristics of the daily load profiles (Figure C.1 in AppendixC) which present three important ranges: a low power demand range during the night, between 23 and 6 hours and two high ranges of load taking place in the intervals 10 to 13 and 18 to 21 hours, respectively. Thus, the left peak of the distributions corresponds to the highest range of loads (18 to 21 hours), because higher levels of power demands imply more energy sold and, therefore, more profits or negative values of CG. Following the same logic, the central peak is due to the second highest range of loads (10 to 13 hours) and the right peak to the low range of power demand. As mentioned before, the three DG–integrated network cases improve the cost profiles because the usage of power is transferred from the bulk–power suppliers (MG) to the renewable generators (RG) as summarized in Table 5.6, presenting the ratio of power usage defined as the proportion of power used to satisfy the loads, determined by the MCS–OPF, over the power available. According to the

IEEE 30 bus test systems information, the bulk–power supply arriving at node 1, G1, comes from a conventional coal–fired power plant whereas G2 is supplied by a hydro–power plant. The nature of the source of power supply conditions the operating costs, in particular, G1 that presents the higher variable operating costs (Table C.3 in AppendixC). Nevertheless, the ratio of power usage associated to G2 is in all cases 100%, even though its operating cost is higher than the renewable.

This is because no investment is being paid for G2, contrary to the DG technologies (PV and W). In this view, considering investment, PV and W are more convenient than the coal–fired power supply

G1 but not more than the hydro–power supply G2.

Table 5.6 Ratio of power usage by type of generator

Case G1 G2 PVW MG 77.93% 100.00% − − ECGmin 41.88% 100.00% 99.96% 99.94% CVaRCGmin 40.38% 100.00% 99.95% 99.42% DCVaRCGmin 40.71% 100.00% 100.00% 98.73%

The extreme CG scenarios encountered in the tail of the distributions are mainly produced by the occurrence of failures in the components of the system, power generators and T&D lines. In Figure 5.15, the stability of the CG to these non-desirable events can be pointed out. Since the cost objective function to be minimized in the OPF considers a load shedding cost, the occurrence of failures in the components, interrupting the power supply and/or the ability of distribute it, will impact the CG function depending on how much centralized is the power supply. Focusing on the MG base case, the power supply and its distribution depend on two generators and the T&D lines connected to them: the reliability of power supply is determined by few components and so, the eventual losses of functionality of these components can lead to high amounts of non–satisfied demands. This is precisely the effect observed in the tails of the CG distributions and, in particular, it is seen that distributing and diversifying generation helps to improve the risk impacts from multiple 5 Applications 71 failures in the network, even if the number of generation units in the system is increased and with it, the overall absolute likelihood of occurrence of failures.

5.3.3 Optimal DG–integrated network plans

Figure 5.16 shows the total average DG power integrated in the network for the three solutions under analysis and the corresponding obtained values of ECG and CVaRα. It can be pointed out that in all cases the DG power installed is almost equal to the limit value set by the penetration DG factor PF = 0.3, approximately 135.5 (MW).

300 50 P (MW) Case [QR ]min AVij, ECG 40 PV 30 G1 W 20 G2 10

0 0 300 50 P (MW) Case [QR ]min AVij, CVaR() CG 40 PV 30 G1 W 20 G2 10

0 0 300 50 P (MW) Case [QR ]min AVij, DCVaR() CG 40 120 -120 PV 30 G1 W 80 20-80 G2 10 40 -40 0 0 0 0

-40 40

-80 80 L (MW) AVi -120 120 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 node i

Figure 5.16 Total average renewable power installed. Cases ECGmin (A), CVaRCGmin (B) and DCVaRCGmin (C)

Furthermore, in general terms moving along the non–dominated solutions, starting from the one R min that minimizes ECG ([Q ]ECG), passing to the compromising one that minimizes CVaRα(CG) ( QR min ) and ending with ( QR min ) that minimizes DCVaR CG , the amount of PV [ ]CVaR(CG) [ ]DCVaR(CG) α( ) 72 5 Applications power integrated in the network decreases progressively, being replaced by W power. Then, it can be inferred that the more PV power generators are integrated, the better ECG performance is achieved. This seems physically coherent, taking into account that the average power outputs delivered by one generation unit of PV and W technologies are 1.07 and 0.93 (MW), respectively, and the cost performance benefits comparatively more from the PV MW h sold during the daylight interval within the two peaks ranges of power demand. Nevertheless, the amount of average integrated PV power is invariably smaller than W power, this is because PV generation units do not supply during the night interval, making convenient to integrate always a certain amount of W power and, thus, to avoid resorting to the more expensive coal–fired power supplier G1. Moreover, it is precisely this lack of PV generation during the night interval that strongly conditions the trade–off between expected performance and uncertainty, ECG vs DCVaRα(CG), i.e., the more PV power is integrated, the better ECG performance is achieved but the more uncertain is its estimation.

PV W

ECG CVaRa (CG) 150 -4.0

100 -5.0 ) G

C (MW) (

, ij a (k$/h)

R AV

a P V

50 C -11.5

D 

0 -12.5 A B C

Figure 5.17 Nodal average power by type of generator

Concerning the compromising solution QR min that minimizes risk, as it was derived from [ ]CVaR(CG) the empirical CG distributions, the risk associated to a solution depends mainly to the occurrence of extreme non-desired events. Improvements in risk performance can, then, be achieved by solutions for which the allocation of DG power generation units decentralizes and diversifies to a large extent the supply. This insight is noticeable in Figure 5.17 that reports the nodal average power by type of R min R min generation for the three solutions of interest. For both extreme solutions, [Q ]ECG and [Q ]DCVaR, the tendency is to integrate localized sources of renewable DG at two identifiable portions of the network, in the region close to nodes 2, 5, 7 and 8 of the sub–transmission portion of the network, and nodes 17, 19 and 21 in the distribution part, favoring the sub–transmission portion which presents higher and non homogeneous nodal load profiles. In a different manner, the solution QR min [ ]CVaR(CG) presents a more homogeneous deployment of DG power, allocating comparable generation capacities in both sub–transmission and distribution parts of the network. 5 Applications 73

5.3.4 Brief summary

We have presented a MOO framework for the integration of renewable distributed generation into an electric power network. Multiple uncertain operational inputs are taken into consideration: the inherent uncertain behavior of renewable energy sources and power demands, as well as the occurrence of failures of components. For managing the uncertainty and risk associated to the achievement of a certain level of expected global cost performance, we have introduced the conditional-value-at-risk deviation measure, which allows trading off the level of uncertainty and, given the axiomatic relation to the conditional-value-at-risk, enables conjointly the trade-off of risk by constructing an iso–risk map in the non–dominated set of solutions. The proposed framework integrates the multi-objective differential evolution as a search engine, Monte Carlo simulation to randomly generate realizations of the uncertain operational scenarios and optimal power flow to evaluate the network response. The optimization is done to simultaneously minimize the expected value of the global costs and the respective conditional–value–at–risk deviation.

A case study has been analyzed, based on the IEEE 30 bus sub–transmission and distribution test system. The results obtained show the capability of the framework to identify Pareto optimal solutions of renewable DG units allocations. Integrating the conditional value–at–risk deviation into the framework has shown effectively the possibility of optimizing expected performances while controlling the uncertainty and risk, analyzing, in addition, the contribution of each type of renewable DG technology on the level of uncertainty associated to the outcome performance of the optimal solutions and the importance of the deployment of the renewable generation capacity to lower the risk of incurring in non-desirable extreme scenarios. In this view, a complete and comprehensible spectrum of information can be supplied in support of specific preferences of the decision makers for their decision tasks.

6 Conclusions

In the present thesis, a modeling, simulation and optimization framework has been designed and developed, for the integration of renewable DG into electric power networks. The DG planning problem considered has been that of optimal selection of the technology, size and location of multiple DG units, subject to technical, operational and economic constraints. Key research questions addressed have been:

(i) Representation and treatment of the multiple uncertain operational inputs such as: the inherent variability in the availability of diverse primary renewable energy sources, bulk– power supply, power demands and components operating states.

(ii) Propagation of the uncertainties onto the system operational response, and control of the associated risk of incurring in extreme non–satisfactory performances.

(iii) Computational efforts resulting from the complex combinatorial optimization problem under uncertainty associated to renewable DG integration.

6.1 Original contributions

The original contributions of the work reside in:

(a) Design and implementation of a non–sequential Monte Carlo simulation (MCS) and optimal power flow (OPF) computational model, denoted MCS–OPF, that emulates the T&D network operation integrating a given renewable DG plan. Random realizations of operational scenarios are generated by latin hypercube (LHC) sampling from the different uncertain variables models, solving for each scenario a cost–based DC–OPF problem to assess the response of the DG–integrated network in terms of, available power usage, power demand satisfaction and operating cost (generation and T&D) to , then, evaluate the performance of the system with regards to the global economics, including DG investments and operation, and reliability of power supply, represented by the global cost (CG) and the energy not supplied (ENS), respectively. The simplifications involving the non–sequential character of the MCS andthe use of DC–OPF approximation are done to not overly increasing the computational efforts and 76 6 Conclusions

gaining tractability, necessary to cope with the already complex DG planning optimization problem.

(b) With respect to the optimal technology selection, size and location of the renewable DG units, two distinct multi–objective optimization (MOO) strategies have been implemented by heuristic optimization (HO) search engines, in which the MCS–OPF model is nested to assess the performance of each DG–integrated network proposed along the evolutionary searching process and the values of the respective objective functions. Two HO search engines have been considered, the fast non–dominated sorting genetic algorithm II (NSGA–II) and a MOO differential evolution (MOO–DE), both capable of dealing with non-convex combinatorial problems, discontinuous search spaces and non-differentiable objective functions.

(c) Introduction of two indicators to measure and control the uncertainty and risk associated to the DG–integrated network solutions, namely conditional value–at–risk (CVaR) and conditional value–at–risk deviation (DCVaR), respectively. This is done, framing the DG planning problem as a portfolio optimization in which the different types of DG technologies are treated analogously to financial assets.

By introducing CVaR, a direct risk-based MOO strategy is formulated and approached ◦ by the NSGA–II, allowing the possibility of concurrently minimizing expected performances, ECG and EENS, while controlling the risk in its achievement, CVaR(CG) and CVaR(ENS). The contribution of each type of renewable DG technology, on the expectation–risk trade–off, can also be analyzed, indicating which is more suitable for specific preferences of the decision makers. Alternatively, DCVaR allows the development of a distinct MOO optimization strategy, ◦ aiming at the simultaneous minimization of the considered objective functions: ECG and its corresponding DCVaR(CG) value and implemented by the MOO–DE search. DCVaR acts as an enabler of trade-off between optimal expected performance and the associated uncertainty to achieve it and, given the axiomatic relation to the conditional–value–at– risk, allows the conjoint trade–off of risk by constructing an iso-risk map in the obtained non-dominated set of solutions. In addition, the contribution of each type of renewable DG technology on the level of uncertainty associated to the outcome performance of the optimal solutions can be analyzed, as well as the importance of the deployment of the DG capacity to lower the risk of incurring in non-desirable extreme scenarios. In this view, a complete and comprehensible spectrum of information can be supplied in support of the decision making tasks.

performing MCS–OPF on selected representative individuals of the groups only, thus reducing the number of objective function evaluations in each iteration of the DE evolution loop. The introduction of two control parameters, namely the cophenetic correlation coefficient CCC

and a cutoff level coefficient of the linkage distances pco, allow the controlled adaptation during the search process and decision on whether or not to perform clustering and at which level of the hierarchical structure. The HCDE framework is capable of identifying optimal plans of renewable DG integration, leading to acceptable reductions in the number of objective function evaluations with small dispersion and loss of quality in the minimum ECG obtained.

6.2 Future work

Further developments can be thought of in all three main areas of focus of this thesis: modeling, simulation and optimization for the DG planning problem. Regarding the modeling of the distribution network and, in particular, the technical constraints imposed by power flow equations, more realistic assumptions can be considered, including the impact of DG in the reactive power balance of the network and, therefore, the voltage and power losses profiles, that are regulated and ought tobe strictly controlled in electric power systems. Even if, at small scales no reactive power requirements are set to renewable power generators [25], high penetration levels of DG integration leads to the need of evaluating the potential contribution of renewable generation to power system voltage and reactive power regulation [105]. This eventually implies to use the AC power flow equations. Another improvement in the modeling of a DG–integrated network is the representation of the evolution of the uncertain operational conditions, such as solar irradiation, wind speed, memory effect of batteries, load profiles, energy prices, etc. The forecast of these variables implies the prediction of future conditions given specific previous scenarios. Then, a sequential MCS model should be developed, in which the sampling is performed for each time step dependent on previous states conditions. As an example, load forecast uncertainty can be integrated properly building consecutive load scenarios and assigning corresponding probabilities of occurrence as presented by [16, 34]. Another interesting approach for load forecast uncertainty modeling is the geometric Brownian motion (GBM) stochastic process [42, 76]. The above–mentioned developments entail an increase in the computational efforts involved in the resolution of the DG planning. Hence, the motivation to explore and develop new methodologies to improve the computational performance still remain. In particular, a direct next step would be the extension of the HCDE search engine into MOO strategies by new controllable clustering techniques, measuring the variation in the quality of non–dominated solutions sets, e.g. by a hypervolume indicator [95].

A IEEE 13–bus test feeder data: Application 5.1

Table A.1 contains the technical characteristics of the different types of feeders considered: specifically, the indexes of the pairs of nodes that are connected by each feeder of the network, their length l, reactance X and their ampacity A.

Table A.1 Feeders characteristic and technical data [2]

Type Node i Node i' l (km) X (Ω/km) A (A) T1 1 2 0.61 0.37 365 T2 2 3 0.15 0.47 170 T3 2 4 0.15 0.56 115 T1 2 6 0.61 0.37 365 T3 4 5 0.09 0.56 115 T6 6 7 0.15 0.25 165 T4 6 8 0.09 0.56 115 T1 6 11 0.31 0.37 365 T5 8 9 0.09 0.56 115 T7 8 10 0.24 0.32 115

The nodal power demands are reported as daily profiles, normally distributed on each hour. The mean µ and variance σ values of the nodal daily profiles of the power demands are shown in Figure A.1(A) and (B), respectively.

1400 20 A B

1050 15

kW)

( (kW)

700 i,t 10

σ μ 350 5

0 0 1 1 3 5 3 5 7 7 9 9 11 11 13 13 t (h) 15 15 17 t (h) 17 19 19 21 21 node i 23 node i 23

Figure A.1 Mean (A) and variance (B) values of nodal power demand daily profiles 80 A IEEE 13–bus test feeder data: Application 5.1

The maximum active power capacity of the transformer (MG) and the parameters of the normal distribution that describe its variability are given in Table A.2.

Table A.2 Bulk–power supply parameters

Normal distribution parameters Node i Pmax (kW) µ (kW) σ (kW) 1 1600 1200 27.5

The technical parameters of the four different types of DG technologies considered to be integrated into the distribution network (PV,W, EV and ST) are given in Table A.3. The values of the parameters of the Beta and Weibull distributions describing the variability of the solar irradiation and wind speed, are assumed constant in the whole network, i.e., the region of distribution is such that the weather conditions are the same for all nodes.

Table A.3 Parameters of PV,W, EV and ST technologies [3–5]

PV W Beta distribution α 0.26 Weibull distribution α 11.25 Beta distribution β 0.73 Weibull distribution β 2

Pmax (W) 50 PR (kW) 50

TA (◦C) 30 UCI (m/s) 3.8

TN o (◦C) 43 UA (m/s) 9.5

ISC (A) 1.8 UCO (m/s) 23.8

kI (mA/◦C) 1.4 EV

VOC (V) 55.5 PR (kW) 6.3

kV (mV/◦C) 194 ST

VMPP (V) 38 PR (kW) 0.275

IMPP (A) 1.32 JS (kJ/kg) 0.042

Failures and repair rates of the components of the distribution network are provided in Table A.4.

Table A.4 Failure rates of feeders, MG and DG units [3–6]

Type λF (n/h) λR (n/h) MG DGFDMG DGTDMG DGFD ∪ ∪ ∪ MG T1 3.33e 04 3.33e 04 0.021 0.198 − − PV T2 4.05e 04 4.05e 04 0.013 0.162 − − W T3 3.55e 04 3.55e 04 0.015 0.185 − − EV T4 3.55e 04 3.55e 04 0.105 0.185 − − ST T5 3.55e 04 3.55e 04 0.073 0.185 − − - T6 - 4.00e 04 - 0.164 − - T7 - 3.55e 04 - 0.185 −

15 10

A IEEE 13–bus test feeder data: Application19 5.1 81

13 8

The hourly per day operating states probability17 profile of the EV is presented in Figure A.2.

11 6

1.0 15



 

0.7 9

ρ0

ρ1

8 3

0.4 12

j ,t j ,t





discharging disconnected

charging 0.1 6

0.6 1

 

0.4 

ρ0

ρ1

0.00

0.25

0.50 0.75

1.00 0.2

j ,t j ,t





discharging

disconnected charging

0.0 6

0.3



 

0.2 0

ρ0

ρ1

0.1 3

j ,t j ,t

0.00

0.25

0.50

0.75

1.00





discharging disconnected charging 0.0

0 2 4 6 1 8 10 12 14 16 18 20 22

0 t (h)

0.00

0.25

0.50 0.75 Figure A.2 1.00 Hourly per day probability data of EV operating states. ρ = 1: charging, ρ = 0: disconnected, ρ = 1: discharging −

The values of the investment (CI) and fixed and variable Operational and Maintenance (CO f and COv) costs of the MG and DG units are reported in Table A.5.

Table A.5 Investment, fixed O&M and variable O&M costs of MG andDG [6–8]

Type CI + CO f ($) COv ($/kWh) MG - 1.45e 01 − PV 48 3.76e 05 − W 113750 3.90e 02 − EV 17000 2.20e 02 − ST 135.15 4.62e 05 −

The total investment associated to DG-integrated network plan must be less than or equal to the limit budget; which is set to BGT = 4500000 ($), and the total number of units of each type of DG (following the order [PV,W, EV,ST]) must be less than or equal to τj = [15000, 5, 200, 8000]. The value of the incentive for renewable kW h supplied is taken as 0.024 ($/kWh) [8]. The maximum value of the energy price EPmax is 0.11 ($/kWh) [52, 53].

Concerning the calculation of the CVaR, the α–percentile is taken as α = 0.80.

B IEEE 13–bus test feeder data: Application 5.2

Table B.1 contains the technical characteristics of the different types of feeders considered: specifically, the indexes of the pairs of nodes (i, i') that they connect, their length l, reactance X , ampacity A and failure and repair rates.

Table B.1 Feeders characteristic and technical data [2,4,9 ]

Type Node i Node i' l (km) X (Ω/km) A (A) λF (n/h) λR (n/h) COv ($/kWh) T1 1 2 0.610 0.371 730 3.333e 04 0.198 1.970e 02 T2 2 3 0.152 0.472 340 4.050e−04 0.162 9.173e−03 T3 2 4 0.152 0.555 230 3.552e−04 0.185 6.205e−03 T1 2 6 0.610 0.371 730 3.333e−04 0.198 6.205e−03 T3 4 5 0.091 0.555 230 3.552e−04 0.185 6.205e−03 T6 6 7 0.152 0.252 329 4.048e−04 0.164 8.904e−03 T4 6 8 0.091 0.555 230 3.552e−04 0.185 1.970e−02 T1 6 11 0.305 0.371 730 3.333e−04 0.198 1.970e−02 T5 8 9 0.091 0.555 230 3.552e−04 0.185 9.173e−03 T7 8 10 0.244 0.318 175 3.552e−04 0.185 6.205e−03 − −

The nodal power demands are built from the load data given in [2] and reported in Figure B.1 as daily profiles, normally distributed on each hour t with mean µ and standard deviation σ [10, 58].

1800 180

node 2 1350 135node 2 node 3

) node 3 node 4

node 4 (kW 900 (kW) 90 node 5

i,t node 5 i,t

μ node 6 σ node 6 node 7 450 45 node 7 node 9 node 9 node 10 0 node0 10 0 4 8 12 16 20 24 0 4 8 12 16 20 24 t (h) t (h)

Figure B.1 Mean and standard deviation values of normally distributed nodal power demand daily profile 84 B IEEE 13–bus test feeder data: Application 5.2

The technical parameters, failure and repair rates and costs of the MG and the four different types of DG technologies (PV,W, EV and ST) available to be integrated into the distribution network are given in Table B.2. For the corresponding application example, the distribution region is such that the solar irradiation and wind speed conditions are assumed uniform in the whole network, i.e., the values of the parameters of the corresponding Beta and Weibull distributions are assumed constant in the whole network.

Table B.2 Power sources parameters and technical data [3–8, 10]

Distributions parameters, Type j Technical parameters Costs failure and repair rates µ = 4000 (kW) σ = 125 (kW) MG Pmax = 4250 (kW) COv = 0.145 ($/kWh) λF = 4.00e 04 (1/h) − λR = 1.30e 02 (1/h) − TA = 30.00 (◦C)

TN o = 43.00 (◦C) ISC = 1.80 (A) α = 0.26 V 55.50 (V) β 0.73 CI CO f 48 ($) PV OC = = + = kI = 1.40 (mA/◦C) λF = 5.00e 04 (1/h) COv = 3.76e 05 ($/kWh) − − kV = 194.00 (mV/◦C) λR = 1.30e 02 (1/h) − VMPP = 38.00 (V) IMPP = 1.32 (A)

UCI+CO f = 3.80 (m/s) α = 11.25 U 9.50 (m s) β 2 CI CO f 113, 750 ($) W A = / = + = UCO = 23.80 (m/s) λF = 6.00e 04 (1/h) COv = 3.90e 02 ($/kWh) − − PR = 50.00 (kW) λR = 1.30e 02 (1/h) − λF = 2.00e 04 (1/h) CI + CO f = 17000 ($) EV PR = 6.30 (kW) − λR = 9.70e 02 (1/h) COv = 2.20e 02 ($/kWh) − − PR = 0.28 (kW/kg) λF = 3.00e 04 (1/h) CI + CO f = 135.15 ($) ST − JS = 0.04 (kJ/kg) λR = 7.30e 02 (1/h) COv = 4.62e 05 ($/kWh) − −

The hourly per day operating state probability profiles of the EV are presented in Figure B.2: p0, + p− and p correspond to the profiles of disconnected, charging and discharging states, respectively.

p0 p− p+

t (h)

Figure B.2 Hourly per day probability data of EV operating states B IEEE 13–bus test feeder data: Application 5.2 85

The budget is set to BGT = 4500000 ($) and the limit of units of the different DG technologies available to be purchased is τ = [20000, 8, 250, 10000]. The maximum value of the energy price is EP 0.12 ($ kWh) 53 and the highest value of total demand ΣL is set to 4800 (kW). max = / [ ] maxi The opportunity cost for kW h not supplied CLS is considered as twice of the maximum energy price.

C IEEE 30–bus sub–transmission & distribution system data: Application 5.3

Table C.1 summarizes the characteristics and technical data of the T&D lines, specifically: the indexes of the pair of nodes that they connect i, i , the susceptance values B i,i , power rating ( ′) ( ′) Pmax , failure λF and repair λR rates and operating cost COv i,i . (i,i ) (i,i ) (i,i ) ( ) ′ ′ ′ ′ The nodal power demands are built from the load data given in [11] and reported in Figure C.1 as daily profiles (accumulated according to the node indexes i), normally distributed on each hour t with mean µi,t and standard deviation σi,t .

350 35 ABnode 30 node 29 300 30 node 26 node 24 node 23 250 25 node 21 node 20 node 19 (MW) (MW) 200 20 node 18 node 17 node 16 node 15 150 15 node 14 c node 12 it, node 10 100 10 i1 node 8 it,,c it i *, t node 7 i*1 node 5 50 i1 5 node 4  it*, i*1 node 3 node 2 0 0 1 3 5 7 9 11 13 15 17 19 21 23 1 3 5 7 9 11 13 15 17 19 21 23 tt

Figure C.1 Accumulated mean (A) and standard deviation (B) values of nodal load daily profiles 88 C IEEE 30–bus sub–transmission & distribution system data: Application 5.3

Table C.1 T&D lines characteristics and technical data [11–13]

i i B i,i (p.u.) Pmax (p.u.) λF (n h) λR (n h) COv i,i ($ MWh) ′ ( ) (i,i ) (i,i ) / (i,i ) / ( ) / ′ ′ ′ ′ ′ 1 2 17.24 1.30 2.85e 04 6.67e 02 4.95 1 3 5.41 1.30 2.85e−04 6.67e−02 1.61 2 4 5.75 0.65 4.28e−04 1.00e−01 2.61 3 4 26.32 1.30 2.85e−04 6.67e−02 5.97 2 5 5.05 1.30 2.85e−04 6.67e−02 2.53 2 6 5.68 0.65 4.28e−04 1.00e−01 3.68 4 6 24.39 0.90 3.73e−04 8.72e−02 6.06 5 7 8.62 0.70 4.17e−04 9.74e−02 0.70 6 7 12.20 1.30 2.85e−04 6.67e−02 0.93 6 8 23.81 0.32 5.01e−04 1.17e−01 0.19 6 9 4.81 0.65 4.28e−04 1.00e−01 5.21 6 10 1.80 0.32 5.01e−04 1.17e−01 4.14 9 10 9.09 0.65 4.28e−04 1.00e−01 8.78 9 11 4.81 0.65 4.28e−04 1.00e−01 4.81 4 12 3.91 0.65 4.28e−04 1.00e−01 2.93 12 13 7.14 0.65 4.28e−04 1.00e−01 1.41 12 14 3.91 0.32 5.01e−04 1.17e−01 11.10 12 15 7.69 0.32 5.01e−04 1.17e−01 6.66 14 15 5.00 0.16 5.36e−04 1.25e−01 13.70 12 16 5.03 0.32 5.01e−04 1.17e−01 13.70 10 17 11.76 0.32 5.01e−04 1.17e−01 13.20 16 17 5.18 0.16 5.36e−04 1.25e−01 8.71 15 18 4.57 0.16 5.36e−04 1.25e−01 3.46 18 19 7.75 0.16 5.36e−04 1.25e−01 12.20 10 20 4.78 0.32 5.01e−04 1.17e−01 5.40 19 20 14.71 0.32 5.01e−04 1.17e−01 3.52 10 21 13.33 0.32 5.01e−04 1.17e−01 11.50 10 22 6.67 0.32 5.01e−04 1.17e−01 9.76 21 22 41.67 0.32 5.01e−04 1.17e−01 8.08 15 23 4.95 0.16 5.36e−04 1.25e−01 1.68 22 24 5.59 0.16 5.36e−04 1.25e−01 7.82 23 24 3.70 0.16 5.36e−04 1.25e−01 16.40 24 25 3.04 0.16 5.36e−04 1.25e−01 7.98 25 26 2.63 0.16 5.36e−04 1.25e−01 9.13 25 27 4.78 0.16 5.36e−04 1.25e−01 5.56 6 28 16.67 0.32 5.01e−04 1.17e−01 1.15 8 28 5.00 0.32 5.01e−04 1.17e−01 0.36 27 28 2.53 0.65 4.28e−04 1.00e−01 6.17 27 29 2.41 0.16 5.36e−04 1.25e−01 14.40 27 30 1.66 0.16 5.36e−04 1.25e−01 14.00 29 30 2.21 0.16 5.36e−04 1.25e−01 11.40 − − C IEEE 30–bus sub–transmission & distribution system data: Application 5.3 89

The technical data and uncertain model parameters of the different types of bulk-power suppliers and DG technologies, available to be integrated into the network, are given in Table C.2. The number of photovoltaic cells per PV generation unit is nc = 20000 and that the region covered by the system is such that the solar irradiation and wind speed conditions are uniform in the whole region, i.e., the values of the parameters of the corresponding Beta and Weibull distributions are taken equal DG for all nodes. The renewable power penetration factor is set to PF = 0.3.

Table C.2 Power generators technical data and uncertain model parameters [3–5,7, 10, 14 ]

Type Technical parameters Distribution parameters

Pmax1 (MW) Normal µ1 Normal σ1 G1 340 300 18.25 MG

Pmax2 (MW) Normal µ2 Normal σ2 G2 50 42.5 5

PAV (MW) Pmax (W) VOC (V) ISC (A) VMPP (V) Beta α Beta β 1.07 75 21.98 5.32 17.32 PV I (A) k (mV C) k (mA C) T ( C) T ( C) 0.50 0.33 RG MPP V /◦ I /◦ N o ◦ A ◦ 4.76 14.4 1.22 43 30 P (MW) P (MW) U (m s) U (m s) U (m s) Weibull α Weibull β W AV R CI / A / CO / 0.93 1.5 5 15 25 15 2.2

Table C.3 reports the failure and repair rates, λF and λR, respectively, the investment and operating costs CI + CO f and the variable operating cost of the different types of power generators.

Table C.3 Power generators failure and repair rates and costs [3,4,7, 10, 14, 15 ]

Type λF (n/h) λR (n/h) CI + CO f (M$/u) COv ($/MWh)

G1 5.13e 04 2.77e 02 29.32 MG − − − G2 6.84e 04 4.16e 02 8.92 − − − PV 6.27e 04 1.30e 02 2.20 9.69 RG − − W 3.42e 04 9.00e 03 1.85 11.05 − −

The maximum value of the energy price is EPmax = 100 ($/MWh) [4, 52, 53] and the corresponding highest value of total demand ΣLmaxi (MW) is set to 445 (MW). The load shedding cost CLS ($/MWh) is considered as the maximum energy price. The horizon of analysis or lifetime of the project is 30 years, in which the investment and operating costs are hourly prorated. The confidence level or α-percentile considered to estimate the values CVaRα and DCVaRα is 75%, arbitrarily chosen.

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Paper (i)

R. Mena, M. Hennebel, Y.–F. Li, C. Ruiz, and E. Zio, “A risk-based simulation and multi–objective optimization framework for the integration of distributed renewable generation and storage,” Renewable and Sus- tainable Energy Reviews, vol. 37, pp. 778-–793, 2014. 102

A Risk-Based Simulation and Multi-objective Optimization Framework for the Integration of Distributed Renewable Generation and Storage

Rodrigo Menaa, Martin Hennebelb, Yan-Fu Lia, Carlos Ruizc, Enrico Zioa,d,1

aChair on Systems Science and the Energetic Challenge, Fondation Electricité de France at CentraleSupeléc, Chatenay-Malabryˆ Cedex, France bCentraleSupeléc, Department of Power & Energy Systems, Gif-Sur-Yvette, France cUniversidad Carlos III de Madrid, Department of Statistics, Madrid, Spain dPolitecnico di Milano, Energy Department, Milan, Italy

Abstract

We present a simulation and multi–objective optimization framework for the integration of renewable generators and storage devices into an electrical distribution network. The framework searches for the optimal size and location of the distributed renewable generation units (DG). Uncertainties in renewable resources availability, components failure and repair events, loads and grid power supply are incorporated. A Monte Carlo simulation and optimal power ﬂow (MCS–OPF) computational model is used to generate scenarios of the uncertain variables and evaluate the network electric performance. As a response to the need of monitoring and controlling the risk associated to the performance of the optimal DG–integrated network, we introduce the conditional value–at–risk (CVaR) measure into the framework. Multi–objective optimization (MOO) is done with respect to the minimization of the expectations of

the global cost (Cg ) and energy not supplied (ENS) combined with their respective CVaR values. The multi–objective optimization is performed by the fast non–dominated sorting genetic algorithm NSGA–II. For exempliﬁcation, the framework is applied to a distribution network derived from the IEEE 13 nodes test feeder. The results show that the MOO MCS–OPF framework is effective in ﬁnding an optimal DG–integrated network considering multiple sources of uncertainties. In addition, from the perspective of decision making, introducing the CVaR as a measure of risk enables the evaluation of trade–offs between optimal expected performances and risks.

Keywords: Distribution network, renewable distributed generation, renewable energy, uncertainty, conditional value–at–risk, simulation, multi–objective optimization, genetic algorithm

1 Corresponding author. Tel: +33 1 4113 1606; fax: +33 1 4113 1272. E-mail addresses: [email protected], [email protected] (E. Zio).

1 103

1 Introduction

Over the last decade, the global energetic situation has been receiving a progressively greater attention. The adverse environmental effects of fossil fuels, the volatility of the energy market, the growing energy demand and the intensive reliance on centralized bulk–power generation have triggered a re/evolution towards cleaner, safer, diversiﬁed energy sources for reliable and sustainable electric power systems [1–6]. The challenges involved have stimulated both technological development of new equipment and devices, and efﬁciency improvements in design, planning, operation strategies and management across generation, transmission and distribution.

In this paper, we focus on distribution networks and the conceptual and operational transition they are facing. Indeed, the traditional passive operation with unidirectional flow supplied by a centralized generation/transmission system, is evolving towards an active operational setting with integration of distributed generation (DG) and possibly bidirectional power flows [7, 8]. DG is defined as ‘an electric power source connected directly to the distribution network or on the customer site of the meter’ [8–10] and in principle offers important technical and economical benefits. Under the assumption that the distribution network operators have control over the dispatching of the DG power, improvement of the reliability of power supply and reduction of the power losses and voltages drops can be achieved. Indeed, DG allocation on areas close to the customers allows the power flowing through shorter paths, and therefore, decreasing the amount of unsatisfied power demand and enhancing the power and voltage profiles. Thus, the eventual intermittence of the centralized power supply can be smoothed [11]. In addition, the modular structure of the DG technologies implies lower financial risks [12, 13] and thus the investments on the power system can be deferred [1, 3]. Most of the actual DG technologies make use of local renewable energy resources, such as wind power, solar irradiation, hydro–power, etc., which makes them even more attractive in view of the requested environmental sustainability (e.g. the Kyoto Protocol [7, 14, 15]). Given the intermittent character of these energy sources, their implementation needs to be accompanied by efficient energy storage technologies. Attentive DG planning is needed to seize the potential advantages associated to DG integration, taking into account specific technical, operational and economic constraints, sources and loads forecasts and regulations. If the practice of selection, sizing and allocation of the different available technologies is not performed attentively, the installation of multiple renewable DG units could produce serious operational complications, in fact, counteracting the potential benefits. Degradation of control and protection devices, reduction of power quality and reliability on the supply, increment in the voltage instability and all related negative impacts on the costs, could become impediments for integration of DG [1–3, 8, 10, 14, 16–20]. Viewing DG planning as a fundamental baseline of advancement, many efforts have been made to solve the associated problem of DG allocation and sizing. Objective functions considered for

2 104 the optimization are of economic, operational and technical type. Among the first type, cost– based objective functions have been used considering the costs of energy and fuel for generation, investments, operation and maintenance, energy purchase from the transmission system, energy losses, emissions, taxes, incentives, incomes, etc. [1–3, 7, 8, 11, 13, 14, 16–27]. The second type of operational objective functions mainly revolves around indexes such as the contingency load loss index (CLLI) [23], expected value of non–distributed energy cost (ECOST), system average interruption duration index (SAIDI), system average interruption frequency index (SAIFI) [7,16,28], expected energy not supplied (EENS) [28, 29], among others. Regarding the third type of objective functions, technical performance indicators include energy losses [1,30] and total voltage deviation (TVD) [18]. Power Flow (PF) equations are typically solved within the optimization problem to evaluate the objective functions, while respecting constraints and incorporating non–convex and non–linear conditions. Given the complexity of the optimization problem, heuristic optimization techniques belonging to the class of Evolutionary Algorithms (EAs) have been proposed as a most effective way of solution [10], including particle swarm optimization (PSO) [23, 24, 27, 31, 32], differential evolution (DEA) [18] and genetic algorithms (GA) [3, 7, 11, 13, 14, 16, 26, 33, 34]. An additional difficulty associated to the problem is the proper modeling of the uncertainties inherent to the behavior of primary renewable energy sources and the unexpected operating events (failures or stoppages) that can affect the generation units. These uncertainties come on top of those already present in the network, such as intermittence and fluctuation in the main power supply due to unavailability of the transmission system, overloads and interruptions of the power flow in the feeders, failures in the control and protection devices, variability in the power loads and energy prices, etc. These uncertainties are incorporated into the modeling by generating a random set of scenarios by Monte Carlo simulation (MCS); the optimization is, then, executed to obtain the optimal expected or cumulative value(s) of the objective function(s) under the set of scenarios considered [2, 3, 7, 16, 28, 32, 34, 35]. In the search for the optimal DG–integrated network, the use of only mean or cumulative values as objective function(s) of the optimization hinders the possibility of controlling the risk of the optimal solution(s): the optimal DG–integrated network may on average satisfy the performance objectives but be exposed to high–risk scenarios with non–negligible probabilities [1,7,16,24,28,36]. The original contributions of this work reside in: addressing the optimal renewable DG technology selection, sizing and allocation problem within a simulation and multi–objective optimization (MOO) framework that allows for assessing and controlling risk; introducing the conditional value–at– risk (CVaR) as a measure of the risk associated to each objective function of the optimization [37,38]. The main sources of uncertainty are taken into account through the implementation of a MCS and OPF (MCS–OPF) resolution engine nested in a MOO based on NSGA–II [39]. The aim of the MOO is, specifically, the simultaneous minimization of the expected global cost (ECg ) and expected

3 105 energy not supplied (EENS), and corresponding CVaR values. A weighting factor β is introduced to leverage the impact of the CVaR in the search of the ﬁnal Pareto optimal renewable DG integration solutions. The proposed framework provides a new spectrum of information for well–supported decision making enabling the trade–off between optimal expected performance and the associated risk to achieve it.

2 Distributed generation network simulation model

This section introduces the MCS–OPF model, including the deﬁnition of the DG structure and conﬁguration, the presentation of the uncertainty sources and their treatment, the MCS for scenarios generation and the OPF formulation for evaluating the performance of the distribution network, in terms of the objective functions of the MOO problem. The outputs of the MCS–OPF model are the probability density functions of the energy not supplied (ENS) and the global cost (Cg ) of the network, and their respective CVaR values.

2.1 Distributed generation network structure and conﬁguration

Four main classes of components are considered in the distribution network: nodes, feeders, renewable DG units and main power supply spots (MS). The nodes can be understood as ﬁxed spatial locations at which generation units and loads can be allocated. Feeders connect different nodes and through them the power is distributed. Renewable DG units and main power supply spots are power sources; in the case of electric vehicles and storage devices they can also act as loads when they are in charging state. The locations of the main supply spots are ﬁxed. The MOO aims at optimally allocating renewable DG units at the different nodes.

MS 1

5 4 2 3 ~ ~ power renewable generation load DG unit

power flow 98 6 7

10 11 ~

Figure 1: Example of distribution network conﬁguration

4 106

Figure 1 shows an example of conﬁguration of a distribution network adapted from the IEEE 13 nodes test feeder [40], for which the regulator, capacitor, switch and the feeders with length equals to zero are neglected.

Each component in the distribution network has its own features and operating states that determine its performance. Assuming stationary conditions of the operating variables, the network operation is characterized by the location and magnitude of power available, the loads and the mechanical states of the components, because degradation or failures can have a direct impact on the power availability (in the DG units, feeders and/or main supply). The renewable DG technologies considered in this work include solar photovoltaic (PV), wind turbines (W), electric vehicles (EV) and storage devices (batteries) (ST). The power output of each of these technologies is inherently uncertain. PV and W generation are subject to variability through their dependence on environmental conditions, i.e., solar irradiance and wind speed. Dis/connection and dis/charging patterns in EV and ST, respectively, further inﬂuence the uncertainty in the power outputs from the DG units. Also generation and distribution interruptions caused by failures are regarded as signiﬁcant.

The following notation is used for sets and subsets of components in the distribution network: N – set of all nodes; MS – set of all types of main supply power sources; DG – set of all DG technologies; PV – set of all photovoltaic technologies; W – set of all wind technologies; EV – set of all electric vehicle technologies; ST – set of all storage technologies; FD – set of all feeders.

The conﬁgurations of power sources allocated in the network, indicating the size of power capacity and the location, is given in matrix form:

MS1 MSj MSm DG1 DGj DGd

ξξξξξ11,,j,m,m,mj,md 1 1 1 1  1  ξ 1        i Ξ  ξξξξξi ,11 i, j i,m i,m  i ,m j  ξ i ,m  d (1)       node    ξξξξξn,11 n,j n,m n,m n,m j ξ n,m  d

ΞMS ΞDG fixed size and decision matrix of type, size location of MS and location of DG units where Ξ – configuration matrix of type, size and location of the power sources allocated in the distribution network; ΞMS – size and location of main supply, fixed part of the configuration matrix; ΞDG – type size and location of DG units, decision variable part of the configuration matrix; n – number of nodes in the network, N ; m – number of main supply type (transformers), MS ; d – | | | |

5 107 number of DG technologies, DG . | |

ζ number of units of MS type or DG technology j are allocated at node i ξi,j =  0 otherwise (2)  i N, j MS DG, ζ  Z∗ ∀ ∈ ∈ ∪ ∈ Feeders deployment is described by the set of pairs of nodes connected:

FD = (1, 2),..., (i, i') (i, i') N N, (i, i') is a feeder (3) { } ∀ ∈ × MS DG Any configuration Ξ, FD of power sources Ξ = [Ξ Ξ ] and feeders FD of the distribution network are affected{ by uncertainty,} so that the operation| and performance of the distribution network is strongly dependent on the network configuration and scenarios. Furthermore, if the distribution network acts as a ‘price taker’, the variability of the economic conditions, particularly the price of the energy, is also an influencing factor [13, 19, 20]. For these reasons, it is imperative to represent and account for the uncertainties in the optimal allocation results for informed and conscious decision–making.

2.2 Uncertainty Modeling

2.2.1 Photovoltaic generation

PV technology converts the solar irradiance into electrical power through a set of solar cells con- ﬁgured as panels. Commonly, solar irradiance has been modeled using probabilistic distributions, derived from the weather historical data of a particular geographical area. The Beta distribution function [41, 42] is used in this paper:

Γ α β ( + ) s(α 1) 1 s (β 1) s 0, 1 , α , β > 0 Γ α Γ β − ( ) − [ ] i i fpv(s) = ( ) ( ) − ∀ ∈ (4)  0 otherwise  where s – solar irradiance; fpv – beta probability density function; α, β – parameters of the beta probability density function. The parameters of the Beta probability density function can be inferred from the estimated mean µ and standard deviation σ of the random variable s as follows [1]:

µ 1 µ β 1 µ ( + ) 1 (5) = ( ) 2 − σ − µβ α (6) = 1 µ − Besides dependence on solar irradiation, PV depends also on the features of the solar cells that

6 108 constitute the panels and on ambient temperature on site. The power outputs from a single solar cell is obtained from the following equations [41, 42]:

N 20 T T s oT (7) c = a + 0.8− I = s(Isc + ki(Tc 25)) (8) − V = Voc + kv Tc (9)

VMPP IMPP FF = (10) Voc Isc pv P (s) = ncells FFVI (11) where Ta – ambient temperature (◦C); NoT – nominal cell operating temperature (◦C); Tc – cell temperature (◦C); Isc – short circuit current (A); ki – current temperature coefficient (mA/◦C); Voc – open circuit voltage (V); kv – voltage temperature coefficient (mV/◦C); VMPP – voltage at maximum power (V); IMPP – current at maximum power (A); FF – fill factor; ncells – number of photovoltaic pv cells; P (s) – PV power output (W).

2.2.2 Wind generation

Wind generation is obtained from turbine–alternator devices that transform the kinetic energy of the wind into electrical power. The stochastic behavior of the wind speed is commonly represented through probability distribution functions. In particular, the Rayleigh distribution has been found suitable to model the randomness of the wind speed in various conditions [1, 42]:

2ws ws 2 fw(ws) = exp (12) σ − σ where ws – wind speed (m/s); fw – Rayleigh probability density function; σ – scale parameter of the Rayleigh distribution function.

Then, for a given wind speed value, the power output of one wind turbine can be determined as [1, 41, 42]: w ws wsci PRTD − if wsci ws < wsa wsa wsci ≤ Pw ws  w − (13) ( ) = PRTD if wsa ws wsco  ≤ ≤  0 otherwise   where wsci – cut–in wind speed (m/s); wsa – rated wind speed (m/s); wsco – cut–out wind speed w w (m/s); PRTD – rated power (kW); P (ws) – wind power output (kW).

7 109

2.2.3 Electric vehicles

In this work, EV are considered as battery electric vehicles with three possible operating states: charging, discharging (i.e., injecting power into the distribution network) and disconnected [43]. To model their pattern of operation, they are considered as a ‘block group’, aggregating their single operating states into an overall performance. The main reasons for this aggregation are the observed nearly stable daily usage schedule of EV and the need of avoiding the combinatorial explosion of the model [42].

1.00 disconnected charging 0.75 discharging

0.50 probability 0.25

0.00 0246810121416182022

td (h)

Figure 2: Hourly probability distribution of EV operating states per day

The power output of one block of EV is formulated by assigning residence time intervals to each possible operating state and associating them with the percentage of trips that the vehicles perform by hour of a day [43]. This allows approximating the hourly probability distribution of the operating states per day, as shown Figure 2. In a given (random) scenario of operational conditions, the determination of the operating state of a block of EV,of a speciﬁc hour of the day, is sampled randomly from the corresponding probability distribution. Accordingly, the power output for a unit or block group of EV is calculated using the expressions (14) and (15) below:

pdch(td ) if op = discharging

fev(td , op) =  pch(td ) if op = charging (14)   pd td (td ) if op = disconnected  op OPs = charging, discharging, disconnected ∀ ∈ { }

8 110

ev PRTD if op = discharging Pev op ev (15) ( ) =  PRTD if op = charging  −  0 if op = disconnected  t [0, tRop], op OPs = charging, discharging, disconnected ∀ ∈ ∈ { } where td – hour of the day (h); tRop – residence time interval for operating state op (h); fev – ev operating state probability density function; PRTD – rated power (kW).

2.2.4 Storage devices

Analogously to the EV case, storage devices are treated as batteries. In reality, these present two main operating states, charging and discharging [44]. However, for this study the level of charge in the batteries is randomized and the state of discharging is the only one that is allowed. This is done to simplify the behavior of the batteries, making it independent on the previous state of charge. The discharging time interval is assigned according to the relation between the batteries rated power, their energy density and the random level of charge they present. For this, the discharging action is carried out at a rate equal to the rated power. Then, the power output per unit of mass of active chemical in the battery MT is estimated as follows:

1 Qst 0, SE M st [ T ] f Q SE MT ∀ ∈ × (16) st ( ) = ×  0 otherwise  Qst t' Qst (17)  R( ) = s P tRTD st st P (tR) = PRTD tR [0, t'R] (18) ∀ ∈ where Qst – level of charge in the battery (kJ); SE – speciﬁc energy of the active chemical (J/kg);

MT – total mass of the active chemical in the battery (kg); fst – uniform probability density function; st PRTD – rated power (kW); t'R – discharging time interval (h).

2.2.5 Main power supply

The MS spots in the distribution network are the power stations connected to the transmission system. The distribution transformers are located on these spots and provide the voltage level of the customers. The stochasticity of the available main supplies of power is represented following

9 111 normal distributions [10, 45], truncated by the maximum capacity of the transformers.

1 P ms µms φ σms σ−ms Pms 0, Pms ms ms ms [ cap] ms  Pcap µ µ fms(P ) = Φ Φ ∀ ∈ (19)  −ms ms  σ − −σ  0 otherwise   where Pms – available main power supply (kW); µms – Normal distribution mean; σms – Normal ms distribution standard deviation; fms – Normal probability density function; Pcap – maximum capacity of the transformer (kW); φ – standard Normal probability density function; Φ – cumulative distribution function of φ.

2.2.6 Mechanical states of the components

Renewable DG units, MS spots and feeders are subject to wearing and degradation processes. These processes can trigger unexpected events, even failures, interrupting or reducing the speciﬁc functionality of each component. Frequently, the stochastic behavior of failures, repairs and maintenance actions is modeled using Markov models [28,42]. In this work, a two–state model is implemented in which the components can be in the mutually exclusive states: available to operate and under repair (failure state). Assuming the duration of each state as exponentially distributed, the mechanical state of a component can be randomly generated as follows:

1 if the component is available to operate mc = (20)  0 otherwise  component Ξ, FD  ∀ ∈ { } 1 mc λF mcλR f mc ( ) + mc 0, 1 (21) mc( ) = − F R λ + λ ∀ ∈ { } F R where mc – binary mechanical state variable, λ – failure rate (failures/h), λ – repair rate (repairs/h), fmc – mechanical state probability mass function.

2.2.7 Demand of power

Overall demands of power, as well as single load profiles in the nodes of the distribution network, can be obtained as daily load curves in which to each hour corresponds one specific level of load, inferred from historical data [1, 14, 19]. In addition, power demands profiles can be considered uncertain following normal distributions [34]. Within the proposed modeling framework, the nodal demands of power are defined by integrating

10 112 the two models mentioned above, i.e. adopting the general daily load proﬁle and considering the hourly levels of load as normally distributed. Figure 3 schematizes the previous assumption for a generic node i.

fid()t

Ltid() ) (kW) d t ( i L

0 t 23 d td (h)

Figure 3: Daily load proﬁle. Hourly normally distributed load

In this manner, the nodal demand of power is deducted from the overall demand in the network, and modeled as:

1 Li µi(td ) φ − σi(td ) σi(td )  i N, Li [0, ] f L , t µi(td ) ∀ ∈ ∈ ∞ (22) Li ( i d ) =  1 Φ  − −σi(td )  0 otherwise   where td – hour of the day (h),Li – power demand in node i (kW), µi – normal distribution mean of power demand in node i, σi – normal distribution standard deviation of power demand in node i, f – normal probability density function of power demand in node i. Li

2.3 Monte Carlo simulation

Most of the techniques used for evaluating the performance of renewable DG–integrated distribution networks are of two classes: analytical methods and MCS [28]. The implementation of analytical methods is always preferable, in theory, because of the possibility of achieving closed exact solutions, but in practice; it often requires strongly simplifying assumptions that may lead to unrealistic results: power network applications exist but for non–ﬂuctuating or non–intermittent generation and/or load proﬁles, and low dimensionality of the network, gaining traceability with reduced computational efforts [32]. Different, MCS techniques allow considering more realistic models that analytical methods do, because simplifying assumptions are not necessary to solve the model, since de facto the model is not solved but simulated and the quantities of interest are estimated

11 113 from the statistics of the virtual simulation runs [46]. For this reason MCS is quite adequate for application on the analysis of distribution networks with signiﬁcant randomness or variability in the sources of power supply and loads, failure occurrence and strong dependence on the power ﬂows as a consequence of congestion conditions in the feeders, etc. [3, 31, 33, 41, 42, 47]; the price to pay for this is the possibly considerable increment in the use of computational resources, and various methods exist to tackle this problem [46]. Given the multiple sources of uncertainties considered in the proposed framework and the proven advantages of MCS for adequacy assessment of power distribution networks with uncertainties [3,31,33,41,42,47], we adopt a non–sequential MCS to emulate the operation of a distribution network, sampling the uncertain variables without considering their time dependence, so as to reduce the computational problem.

For a given structure and configuration of the distribution network Ξ, FD , i.e., for the fixed ΞMS and FD deployments and the proposed renewable DG integration plan{ denoted} by ΞDG, each uncertain variable is randomly sampled. The set ϑ of sampled variables constitutes an operational scenario, in correspondence of which the distribution network operation is modeled by OPF and its performance evaluated. The two inputs to the OPF model are the network configuration Ξ, FD and the operational conditions scenario ϑ. { }

ms ev st ϑ = [td , Pi,j , Li, si, wsi, opi,j,Qi,j, mci,j, mc(i,i')] i, i' N, j MS DG, (i, i') FD (23) ∀ ∈ ∈ ∪ ∈ where td – hour of the day (h), randomly sampled from a discrete uniform distribution U(1, 24). Figure 4 shows an example of the matrix form construction of the DG–integrated distribution network, considering a simple case of n = 3 nodes. The network contains one MS spot at node MS DG i = 1, defining the fixed part Ξ of the configuration matrix, whereas, the decision variable Ξ proposes a renewable DG integration plan ΞDG that built from the number of units ξ of each DG technology allocated. In this way, the network configuration Ξ, FD is composed by the matrix MS DG { } Ξ = [Ξ Ξ ] and the deployment of feeders. Then, given the spatial representation Ξ, FD , the sampling| of the scenario ϑ determines the operational conditions to perform power flow{ analysis,} ϑ i.e., distribute the power available PGa to supply appropriately the demands Li. The available power in the power source type j at node i, Pϑ , is function of the number of units allocated ξ , the Gai,j i,j mechanical state mci,j and the specific unitary power output function associated to the generation unit j, formulated in equations (24) and (25).

12 114

A Construction of the model j 1 2 3 4 5 i = 1 2 3 MS PV W EV ST 1 2 0 1 0 MS ■  L2 L3 Ξ  0 0 3 2 1  PV ■■ ■ ξ 0 1 2 0 3 W ■■■ ■■ i,j EV ■ ■■ units MS DG ST ■ ■■■ allocated Ξ Ξ FD = {(1, 2),(2, 3)} B Sampling of scenarios ϑ ms; mc P  w  11, 1 3()mcP ws  st st; 23, 3 2 3()mc35,,P5 Q 25

pv  ev   w  2()mcs12, P2 1 2()mc24, P24, op 4 2()mc33, P3 ws 3

 st st;  pv  mc32, P2 ()s 3 ev mc25,,P5 ()Q 25 mopc14,,P4 () 14

P P P P P P P P P Ga11, Ga12, Ga14, Ga23, Ga24, Ga25, Ga32, Ga33, Ga35,

  1 mc(1, 2) 23mc(2, 3)

L3(td) L2(td)

Figure 4: Example of the matrix form construction of a DG–integrated network (A) and schema of the operating state deﬁnition from the sampled variables (B)

Pϑ ξ mcϑ G ϑ (24) Gai,j = i,j i,j j( ) ms;ϑ Pj if j MS pv ϑ ∈  Pj (si ) if j PV  ∈ G ϑ  w ϑ (25) j( ) =  Pj (wsi ) if j W i N  ∈ ∀ ∈  ev ϑ Pj (opi,j) if j EV ∈  Pst Qst;ϑ if j ST  j ( i,j )  ∈  In the proposed non–sequential MCS procedure, the intermittency in the solar irradiation is taken into account defining a night interval between 22.00 and 06.00 hours, i.e., if the value of the hour of the day td (h), sampled from a discrete uniform distribution U(1, 24), falls in the night interval, there is no solar irradiation. Regarding the wind speed, its variability is considered by sampling positive values from a Rayleigh probability density function fitted on historical data and whose parameters as such that the probability of absence of wind is zero. Since it is not reasonable to force the historical profile of the wind speed to follow a distribution that admits intermittency, a common alternative technique is to model the wind by a Markov Chain. Indeed, it is possible to accurately represent the wind speed by a stationary Markov process if the historical profile of wind speed data is sufficiently large e.g. years [28]. The intermittency is, then, represented by the first state of the chain with wind speed equals to zero, and the sampling of the wind speed states in the non–sequential MCS of the proposed framework, can be performed using the steady–state

13 115 probabilities of the Markov Chain.

An important issue in modeling the operation of power systems is how to represent the evolution of uncertain operating conditions, such as solar irradiation, wind speed, load proﬁles, energy prices, among others. As an example, the load forecast implies the prediction of future power demands given speciﬁc previous conditions. Therefore, to consider load forecast uncertainty within the proposed MCS framework, it would be necessary to change to a sequential simulation model, in which the uncertain renewable energy resources, main power supply and loads must be sampled at each time step. In particular, load forecast uncertainty can be integrated properly building consecutive load scenarios and assigning corresponding probabilities of occurrence as presented by [7] and [48]. Another interesting approach for load forecast uncertainty modelling is the geometric Brownian motion (GBM) stochastic process [31, 49].

2.4 Optimal power ﬂow

Power flow analysis is performed by DC OPF [50] which takes into account the active power flows, neglecting power losses, and assumes a constant value of the voltage throughout the network. This allows transforming to linear the classic non–linear power flow formulation, gaining simplicity and computational tractability. For this reason, DC power flow is often used in techno–economic analysis of power systems, more frequently in transmission [50, 51] but also in distribution networks [51]. The DC power flow generic formulation is:

Pi = B(i,i')(δi δi') i N, (i, i') FD (26) i' N − ∀ ∈ ∈ X∈ P L P 0 i N (27) Gi i i = i N − − ∀ ∈ X∈ 1 where, Pi – active power leaving node i (kW); B(i,i') – susceptance of the feeder (i, i') (Ω− ); δi – voltage angle at node i; P – active power injected or generated at node i (kW); L – load at node i Gi i (kW).

The assumptions are:

• the difference between voltage angles is small, i.e., sin(∆δ) ∆δ, cos(∆δ) 1 ≈ ≈ • the feeders resistance are neglected, i.e., R << X , which implies that power losses in the feeder are also neglected

• the voltage proﬁle is ﬂat (constant V , set to 1 p.u.)

Then, for a given conﬁguration Ξ, FD and operational scenario ϑ the formulation of the OPF { }

14 116 problem is: net;ϑ ϑ ϑ S min C P C v P t (28) O&M ( Gu) = O&Mj Gui,j i N j MS DG X∈ ∈ X∪ s.t.

ϑ ϑ ϑ ϑ ϑ ϑ L P mc B i,i' δ δ LS 0 i, i' N, i, i' FD (29) i Gui,j (i,i') ( )( i i' ) i = ( ) − j MS DG − i' N − − ∀ ∈ ∈ ∈ X∪ X∈ ϑ ϑ PGu PGa i N, j MS DG (30) i,j ≤ i,j ∀ ∈ ∈ ∪ ϑ 0 PGu i N, j MS DG (31) ≤ i,j ∀ ∈ ∈ ∪ ϑ ϑ ϑ mc i,i' B(i,i')(δi δi' ) V Amp(i,i') i, i' N, (i, i') FD (32) ( ) − ≤ × ∀ ∈ ∈ ϑ ϑ ϑ mc i,i' B(i,i')(δi δi' ) V Amp(i,i') i, i' N, (i, i') FD (33) − ( ) − ≤ × ∀ ∈ ∈ S net;ϑ where, t – duration of the scenario (h); CO&M – operating and maintenance costs of the total power supply and generation ($); C v – operating and maintenance variable costs of the power O&Mj source j ($/kWh); mcϑ – mechanical state of the feeder i, i' ; B – susceptance of the feeder (i,i') ( ) (i,i') i, i' (Ω 1); mcϑ – mechanical state of the power source j at node i; Pϑ – available power in the ( ) − i,j Gai,j source j at node i (kW); Pϑ power produced by source j at node i (kW); LSϑ – load shedding at Gui,j i node i (kW); V – nominal voltage of the network (kV); Amp(i,i') – ampacity of the feeder (i, i') (A).

The load shedding in the node i, LSi, is deﬁned as the amount of load(s) disconnected in node i to alleviate overloaded feeders and/or balance the demand of power with the available power supply [52]. The OPF objective is the minimization of the operating and maintenance costs associated to the generation of power for a given scenario ϑ of duration tS. Equation (29) corresponds to the power balance equation at node i, while equations (30) and (31) are the bounds of the power generation and equations (32) and (33) account for the technical limits of the feeders.

2.5 Performance indicators

Given a set Υ of ns sampled operational scenarios ϑ`, ` 1,..., ns , the OPF is solved for each ∈ { } scenario ϑ` Υ , giving in output the values of ENS and global cost. ∈

2.5.1 Energy not supplied

ENS is a common index for reliability evaluation in power systems [1, 10, 11, 48, 49, 52–55]. In the present work, its value is obtained directly from the OPF output in the form of the aggregation of

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all nodal load sheddings per scenario ϑ`:

ϑ` ϑ` S ENS = LSi t ϑ` Υ (34) i N × ∀ ∈ X∈ Υ ϑ ϑ ϑ ENS = ENS 1 ,..., ENS ` ,..., ENS ns (35) { }

2.5.2 Global cost

The Cg of the distribution network is formed by two terms, fixed and variable costs. The former term includes those costs paid at the beginning of the operation after the installation of the DG (conception of ΞDG). They are the investment–installation cost and the operation–maintenance fixed cost. The variable term refers to the operating and maintenance costs. Note that these costs are dependent on the power generation and supply, which are a direct output of the OPF (eq. (28)). In addition, this term considers revenues associated to the renewable sources incentives. Considering the distribution network as a ‘price taker’ entity, the profits depend on the value of the energy price that is correlated with the total load in the network. Three different ranges of load are considered for the daily profile. For each range, a correlation value of energy price is considered as shown in Figure 5(A).

A 1.00 B high load range LTh ep medium load h 0.75

range LTm epm h ep

/ 0.50 ) (kW) d ep

t low load ($/kWh) (

T range LTl

L 0.25 ep epl 0.00 0.00 0.25 0.50 0.75 1.00 L (t )/L 0 23 T d Th td (h)

Figure 5: Example of load ranges deﬁnition for a generic daily load proﬁle (A) and correlation energy price–total load (B) [13, 19, 20]

In Figure5(B) the correlation between energy price and total load is presented as the proportion of their maximum values. As an intermediate approximation of existing studies (e.g. [13, 19, 20]), the line with square–markers represents the proportional correlation used in this study, which can be expressed as: 2 LT (td ) LT (td ) ep = eph 0.38 + 1.38 (36) − LTh LTh

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Thereby, the global cost function for a scenario ϑ` is given by:

S ϑ t net;ϑ` ϑ ϑ` S C ` C C f C inc ep L ` P t (37) g = ( invj + ) h + O&M ( + ( )) Gu O&Mj t i,j i N j DG − i N j DG X∈ X∈ X∈ X∈ Υ ϑ1 ϑ` ϑns Cg = Cg ,..., Cg ,..., Cg (38) { } where C – investment cost of the DG technology j ($); C f – operating and maintenance ﬁxed invj O&Mj costs of the DG technology j ($); th – horizon of analysis (h); inc – incentive for generation from ϑ` renewable sources ($/kWh); ep – energy price ($/kWh); Cg – global cost ($).

2.5.3 Risk

In [38], the importance of measuring risk when optimizing under uncertainty and including it as part of the objective function(s) or constraints is emphasized. The proposed MOO framework introduces the CVaR as a coherent measure of the risk associated to the objective functions of interest. The CVaR has been broadly used in ﬁnancial portfolio optimization either to reduce or minimize the probability of incurring in large losses [37,38]. This risk measurement allows evaluating how ‘risky’ is the selection of a solution leading to a determined value of expected losses.

We can consider a fixed configuration of the distribution network Ξ, FD including the inte- {Υ } Υ gration of DG units as a portfolio. The assessed expectations of ENS and Cg , found from the Υ MCS–OPF applied to the set of scenarios Υ , are estimations of the losses; then, CVaR(ENS ) and Υ CVaR(Cg ) represent the risk associated to the solutions with these expectations. The definition of CVaR for continuous and discrete general loss functions is given in detail in [38]. Here a simplified and intuitive manner to understand the CVaR definition and its derivation according to [56] is presented.

A B VaR α 1 α+-percentile ……

α-percentile ) losses (  frequency F maximum    loss VaR α 1 VaR α CVaRα 0 α-percentile losses losses

Figure 6: Graphic representation of the CVaR

As shown in Figure 6(A), for a discrete approximation of the probability of the losses, given

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a conﬁdence level or α-percentile, the value–at–risk VaRα represents the smallest value of losses for which the probability that the losses do not exceed the value of VaRα is greater than or equal to α. Thus, from the cumulative distribution function F(losses) is possible to construct the α-tail cumulative distribution function Fα(losses) for the losses, such that (Figure 6(B)):

F losses α ( ) if VaR losses 1 α − α Fα(losses) = ≤ (39)  −0 otherwise 

The α-tail cumulative distribution function represents the risk ‘beyond the VaR’ and its mean value corresponds to the CVaRα. Among other risk measures, the CVaR has been commonly used to assess the ﬁnancial impact associated to different sources of uncertainty on electricity markets behavior. Some interesting approaches in the use of diverse risk measures for electricity markets modelling can be found in [49, 57, 58].

3 DG units selection, sizing and allocation

This section presents the general formulation of the MOO problem considered previously. As introduced, the practical aim of the MOO is to ﬁnd the optimal integration of DG in terms of selection, sizing and allocation of the different renewable generation units (including EV and ST). The corresponding decision variables are contained in ΞDG of the conﬁguration matrix Ξ.

The MOO problem consists in the concurrent minimization of the two objective functions measuring the Cg and ENS, and their associated risk. Speciﬁcally, their expected values and their CVaR values are combined, weighted by a factor β [0, 1], which allows modulating the expected performance of the distribution network and its associated∈ risk.

3.1 MOO problem formulation

Considering a set of randomly generated scenarios Υ , the optimization problem is formulated as follows:

Υ Υ min f1 = βEENS + (1 β)CVaRα(ENS ) (40) − Υ Υ min f2 = βECg + (1 β)CVaRα(Cg ) (41) −

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s.t.

ζ number of units of MS type or DG technology j are allocated at node i ξi,j = (2)  0 otherwise  i N, j MS DG, ζ  Z∗ ∀ ∈ ∈ ∪ ∈ ξ C C f BGT (42) i,j( invj + ) O&Mj i N j DG ≤ X∈ X∈ ξi,j τj j DG (43) i N ≤ ∀ ∈ X∈ OPF( Ξ, FD , Υ ) (28 -33) { } where ECg and EENS denote the expected values of Cg and ENS, respectively.

The meaning of each constraint is, (2) – the decision variable ξi,j is a non–negative integer number; (42) – the total costs of investment and ﬁxed operation and maintenance of the DG units must be less or equal to the available budget BGT; (43) – the total number of DG units to allocate of each technology j must be less or equal to the maximum number of units available τj to be integrated; (28)-(33) – all the equations of OPF must be satisﬁed for all scenarios in Υ .

Constraint (43) can be translated into maximum allowed penetration factor PF DG of each DG max j technology j. Deﬁning PF as ‘the output active power of total capacity of DG divided by the total network load’ [59], constraint (43) can be rewritten as follows:

DG ξi,j EPj DG τj EP i N j j DG (44) P∈ ELT ≤ ELT ∀ ∈ PF DG PF DG j max j

| {z } | {z } DG where i N ξi,j – total number of units of DG technology j integrated in the network; EPj – expected power∈ output of one unit of DG technology j (kW); EL – expected total load (kW). P T The MOO optimization problem is non–linear and non–convex, i.e., a non–convex mixed–integer non–linear problem or non–convex MINLP.It is non–linear because the objective functions given by equations (40) and (41) cannot be written in the canonical form of a linear program, i.e., C T X , where C is a vector of known coefﬁcients and X the decision vector. In the present case, the decision DG matrix Ξ enters the MCS–OPF ﬂow simulation to obtain the probability mass functions of Cg and ENS and, then, the objective functions are formed from the corresponding expected and CVaR values. Thus, the operations applied on ΞDG through MCS–OPF, expectation and CVaR cannot not be represented as the product C T ΞDG. The problem is non–convex because the decision matrices ΞDG are integer–valued (constraint (2)) and, as it is known, the set of non–negative integers is non–convex.

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Given the class of optimization problem in the proposed framework (non–convex MINLP), it is most likely to have multiple local minima. Moreover, the dimension of the distribution network DG can lead to a combinatorial explosion of the feasible space of the decision matrices Ξ [7, 10], incrementing the number of possible local minima and hindering the possibility of benchmarking the optimal solutions obtained. However, an approximated but straightforward alternative is to perform several realizations of the framework obtaining different optimal solutions under the same optimization and simulation conditions (parameters) and, thus, compare them regarding the optimal decision matrices and their associated value of the objective functions.

This process was performed for the proposed case study. Indeed, the optimal decision matrices ΞDG are different in all the cases, when the optimization and simulation framework is performed under the same conditions but, nonetheless, practically the same Pareto optimal values of ECg and EENS are eventually obtained. This reﬂects that equally expected performances (ECg , EENS) can be obtained for different ΞDG considering the large amount of feasible combinations, which is what is of interest for practical applications.

3.1.1 NSGA–II with nested MCS–OPF

The combinatorial MOO problem under uncertainties is solved by the NSGA–II algorithm [39], in which the evaluation of the objective functions is performed by the developed MCS–OPF. The NSGA–II is one of the most efﬁcient evolutionary algorithms to solve MOO problems [60]. The extension to MOO entails the integration of Pareto optimality concepts. In general terms, solving a MOO problem of the form:

min f1(X ), f2(X ),..., fk(X ) X { } (45) s.t. X Λ ∈ n with at least two conﬂicting objectives functions (fi : ) implies to ﬁnd, within a set of acceptable solutions that belong to the non–empty feasibleℜ → region ℜ Λ n, the decision vectors ⊆ ℜ X Λ that satisfy the following [61]: ∈

X Λ : fi(X ) fi(X '), i 1, . . . , k and fi(X ) < fi(X ') for at least one i ¬ ∈ ≤ ∀ ∈ { } (46) ⇓ fi(X ) fi(X ') i.e. X dominates X ' ≺ The vector X is called a Pareto optimal solution and the Pareto front is deﬁned as the set n f (X ) : X is Pareto optimal solution . { ∈ ℜ } The process of searching the non–dominated solutions set, carried out by the NSGA–II MCS–OPF, can be summarized as shown in Figure 7.

The interested reader can consult [62–64] to compare the proposed framework to alternative MOO analytical approaches in energy applications.

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NSGA-II

Set maximum number of generations gen, population size sz and β

chromosome encoding

ξ11, ξ 1 , j ξ 1 ,d Random generation of the initial population p0 of size sz 2D and  DG  integer-valued chromosomes Ξ 0 (constraint (2)) subject to DG Ξ0,k  ξi,1 ξ i, j ξ i,d budget and DG units number constraints, (42) and (43) respectively.  DG DG DG  p0{ΞΞΞ} 0, 1 ,..., 0 ,k ,..., 0 ,sz ξn,1 ξ n, j ξ n,d nodes ofnumber number of DG technologies available d Concatenate each chromosome k of p0 to the fixed configuration matrix of the main supply forming the matrix of power sources MS DG MCS-OPF Ξ 00 ,k  [Ξ | Ξ ,k ] and integrate the feeders set FD to define the

network configuration {Ξ0,k, FD} Set k = 1

Perform the MCS-OPF on p0 Set ℓ = 1

Rank the chromosomes in p by running the fast non-dominated 0 From equations (4) - (22) sample the scenario sorting algorithm with respect to the values objectives functions ms st []td ,P i, j ,L i ,s i ,ws i ,Q i, j ,mc i, j ,mc() i,i' and identify the ranked non-dominated fronts {F1,…,Fnd} where F1 is the best front and Fnd the less good front [39] Solve the OPF problem, equations (26) - (33) MS DG OPF({Ξg ,k,FD } ,| ) with Ξ g ,k [Ξ Ξ g ,k ] Apply the binary tournament selection to p0 based on the crowding distance to generate an intermediate population p 0 of size sz [39] Compute the energy non supplied ENS ENSϑℓ from equation (34) Reproduce the population applying the mutation and crossover operators to p  to create and offspring population o of size sz 0 0 ϑℓ Compute the global costCCg g from equations (36) - (37)

Perform the MCS-OPF on o0 Set ℓ = ℓ +1

Set g = 0 and execute the evolution loop not Is ℓ > ns? yes Combine p and o to obtain a population equals to their union g g Evaluate the objective functions (40) and (41) from the sets of ug p g o g   values of CgCg and ENS obtained from all the simulated scenarios in the set ϒ={ϑ1,…, ϑℓ,…, ϑns}    Perform the MCS-OPF on u ENSf β EENS (1 β) CVaR ( ENS ) g 2 α    Cgf1 β EC g (1 β) CVaRα ( C g )

Rank the chromosomes in ug by running the fast non-dominated sorting algorithm with respect to the values objectives functions Set k = k + 1

not Select the best sz chromosomes to create the next population of parents pg+1 Is k > sz? yes

Apply the binary tournament selection to pg+1 based on the crowding return the values of f1 and f2 for each distance to generate an intermediate population pp’gg+11 of size sz chromosome in the population

Reproduce p’pg+1g 1 applying the mutation and crossover operators to create and offspring population og+1 of size sz

Set g = g + 1

not yes Return p and select the best front F as the Is g > gen? g 1 optimal set of non-dominated solutions

Figure 7: Flow chart of NSGA–II MCS–OPF MOO framework

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4 Case study

We consider a distribution network adapted from the IEEE 13 nodes test feeder [40,65]. The spatial structure of the network has not been altered but we neglect the regulator, capacitor and switch, and remove the feeders of zero length. The network is chosen purposely small, but with all relevant characteristics for the analysis, e.g. comparatively low and high spot and distributed load values and the presence of a power supply spot [65]. The original IEEE 13 nodes test feeder is dimensioned such that the total power demand is satisfied without lines overloading. We modify it so that it becomes of interest to consider the integration of renewable DG units. Specifically, the location and values of some of the load spots and the ampacity values of some feeders have been modified in order to generate conditions of power congestion of the lines, leading to shortages of power supply to specific portions of the network.

4.1 Distribution network description

The distribution network presents a radial structure of n = 11 nodes and f d = (n 1) = 10 feeders, − as shown in Figure 8. The nominal voltage is V = 4.16 (kV), constant for the resolution of the DC optimal power ﬂow problem.

5 4 i = 1 9 8 2 MS 6 10 3 11 7 i: node index MS: Main Supply spot spot load (kW)

Figure 8: Radial 11–nodes distribution network

Table 1: Feeders characteristic and technical data [40]

type node i node i' length (km) X (Ω/km) Amp (A) T1 1 2 0.61 0.37 365 T2 2 3 0.15 0.47 170 T3 2 4 0.15 0.56 115 T1 2 6 0.61 0.37 365 T3 4 5 0.09 0.56 115 T6 6 7 0.15 0.25 165 T4 6 8 0.09 0.56 115 T1 6 11 0.31 0.37 365 T5 8 9 0.09 0.56 115 T7 8 10 0.24 0.32 115

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Table 1 contains the technical characteristics of the different types of feeders considered: specifically, the indexes of the pairs of nodes that are connected by each feeder of the network, their length, reactance X and their ampacity Amp.

Concerning the main power supply spot, the maximum active power capacity of the transformer and the parameters of the normal distribution that describe its variability are given in Table 2.

Table 2: Main power supply parameters

Normal distribution parameters node i P ms (kW) cap µms (kW) σms (kW) 1 1600 1200 27.5

The nodal power demands are reported as daily proﬁles, normally distributed on each hour. The mean µ and variance σ values of the nodal daily proﬁles of the power demands are shown in Figure 9(A) and (B), respectively.

1400 20 AB 1050 15

700 ) (kW) 10 ) (kW) d d t t ( ( i i σ μ 350 5

0 0 1 1 3 5 3 5 7 7 9 9 11 11 13 13 t (h) 15 15 d 17 td (h) 17 19 19 21 21 node i 23 node i 23

Figure 9: Mean (A) and variance (B) values of nodal power demand daily proﬁles

Table 3: Parameters of PV,W, EV and ST technologies [11, 13, 42]

PV W Beta distribution α 0.26 Rayleigh distribution σ 7.96 w Beta distribution β 0.73 PRTD (kW) 50

Ta (◦C) 30 wsci (m/s) 3.8

NoT (◦C) 43 wsa (m/s) 9.5

Isc (A) 1.8 wsco (m/s) 23.8

ki (mA/◦C) 1.4 EV ev Voc (V) 55.5 PRTD (kW) 6.3

kv (mV/◦C) 194 ST st VMPP (V) 38 PRTD (kW) 0.275

IMPP (A) 1.32 SE (kJ/kg) 0.042

The technical parameters of the four different types of DG technologies available to be integrated into the distribution network (PV,W, EV and ST) are given in Table 3. The values of the parameters

23 125 of the Beta and Rayleigh distributions describing the variability of the solar irradiation and wind speed, are assumed constant in the whole network, i.e., the region of distribution is such that the weather conditions are the same for all nodes.

The hourly per day operating states probability proﬁle of the EV is presented in Figure 10 and failures and repair rates of the components of the distribution network are provided in Table 4.

1.0 disconnected 0.7

0.4

0.1 0.6 charging 0.4

0.2 probability 0.0 0.3 discharging 0.2

0.1

0.0 0246810121416182022

td (h)

Figure 10: Hourly per day probability data of EV operating states

Table 4: Failure rates of feeders, MS and DG units [11, 13, 42, 66]

F R type λ (failures/h) λ (repairs/h) MS DGFDMS DGFDMS DGFD MS∪ T1 3.33e∪ 04 3.33e 04 0.021∪ 0.198 PV T2 4.05e−04 4.05e−04 0.013 0.162 W T3 3.55e−04 3.55e−04 0.015 0.185 EV T4 3.55e−04 3.55e−04 0.105 0.185 ST T5 3.55e−04 3.55e−04 0.073 0.185 - T6 -− 4.00e−04 - 0.164 - T7 - 3.55e−04 - 0.185 −

The values of the investment (Cinv) and ﬁxed and variable Operational and Maintenance

(CO&M f and CO&M v ) costs of the MS and DG units are reported in Table 5. Consistently with the constraints (42) and (43) of the MOO problem, the total investment associated to a decision variable ΞDG (proposed by the NSGA–II) must be less than or equal to the limit budget; which is set to BGT = 4500000 ($), and the total number of units of each type of DG (following the order [PV,W,

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EV, ST]) must be less than or equal to τj = [15000, 5, 200, 8000]. The value of the incentive for renewable kWh supplied is taken as 0.024 ($/kWh) [34]. The maximum value of the energy price eph is 0.11 ($/kWh) [19, 20]. Concerning the calculation of the CVaR, the α-percentile is taken as α = 0.80.

Table 5: Investment, ﬁxed O&M and variable O&M costs of MS and DG [27, 34, 66]

type Cinv + CO&M f ($) CO&M v ($/kWh) MS - 1.45e 01 PV 48 3.76e−05 W 113750 3.90e−02 EV 17000 2.20e−02 ST 135.15 4.62e−05 −

Five optimizations runs of the NSGA–II with the nested MCS–OPF algorithm have been performed, each one with a different value of the weight parameter β {1, 0.75, 0.5, 0.25, 0}, to analyze different tradeoffs between optimal average performance and∈ risk. From equations (40) and (41), note that the value β = 1 corresponds to optimizing only the expected values of ENS and Cg , whereas β = 0 corresponds to the opposite extreme case of optimizing only the CVaR values. Each NSGA–II run is set to perform g = 300 generations over a population of sz = 100 chromosomes and, for the reproduction, the single–point crossover and mutation genetic operators are used. The crossover probability is pco = 1, whereas the mutation probability is pmu = 0.1; the mutation can occur simultaneously in any bit of the chromosome.

S Finally, sn = 250 random scenarios are simulated by the MCS–OPF with time step t = 1 (h). h Over an horizon of analysis of 10 years (t = 87600 (h)), in which the investment and ﬁxed costs are prorated hourly.

4.2 Results and discussion

The Pareto fronts resulting from the NSGA–II MCS–OPF are presented in Figure 11 for the different values of β. The ‘last generation’ population is shown and the non–dominated solutions are marked in bold. Each non–dominated solution in the different Pareto fronts corresponds to an optimal decision matrix ΞDG for the sizing and allocation of DG, i.e., an optimal DG–integrated network MS DG conﬁguration Ξ, FD where Ξ = [Ξ Ξ ]. { } | In the Pareto fronts obtained, we look of three representative non–dominated solutions for the DG analysis: those with minimum values of the objective functions f1 and f2 independently (Ξ minf1 and ΞDG , respectively) and an intermediate solution at the ‘elbow’ of the Pareto front. Table 6 minf2 presents the values of the objective functions, EENS, ECg and their respective CVaR values for the selected solutions. The EENS, ECg and CVaR values of the case in which no DG is integrated in the network (MS case) is also reported.

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167.5 β = 1 167.5 β = 0.75 175.0 β = 0.5 ) ($) g ) ($) g C ( C α (

160.0 162.5 α 170.0 ($) CVaR CVaR g EC 0.25 0.5 + +

152.5 g 157.5 g 165.0 EC EC 0.5 0.75 145.0 152.5 160.0 650 725 800 875 750 850 950 1050 800 950 1100 1250

EENS (kWh) 0.75 EENS+0.25 CVaRα(ENS) (kWh) 0.5 EENS+0.5 CVaRα(ENS) (kWh) 180.0 β = 0.25 185.0 β = 0 ) ($) g C ( α 175.0 180.0 ) ($) g CVaR C ( α 0.75 +

g 170.0

CVaR 175.0 EC 0.25

165.0 170.0 1000 1100 1200 1300 1000 1200 1400 1600

0.25 EENS+0.75 CVaRα(ENS) (kWh) CVaRα(ENS) (kWh)

Figure 11: Pareto fronts for different values of β

Table 6: Objective functions: expected and CVaR values of selected Pareto front solutions

β f1 (kW h) f2 ($) EENS (kW h) CVaR(ENS) (kW h) ECg ($) CVaR(Cg ) ($) MS - - - 1109.21 1656.53 170.27 179.24 ΞDG 666.95 160.91 666.95 1093.12 160.91 185.11 minf1 DG 1 Ξel bow 671.05 150.83 671.05 1185.53 150.83 179.47 ΞDG 726.57 148.68 726.57 1279.37 148.68 178.23 minf2 ΞDG 797.07 166.41 677.74 1155.11 160.68 183.62 minf1 DG 0.75 Ξel bow 805.27 159.35 697.17 1129.62 153.09 178.15 ΞDG 867.08 155.61 729.81 1278.94 147.66 179.45 minf2 ΞDG 868.61 171.54 641.68 1095.52 159.43 183.64 minf1 DG 0.5 Ξel bow 936.58 166.67 701.72 1171.47 154.67 178.53 ΞDG 1131.64 162.99 843.53 1419.79 150.45 175.58 minf2 ΞDG 1033.65 172.95 723.19 1137.18 156.55 178.42 minf1 DG 0.25 Ξel bow 1076.53 171.25 743.61 1187.43 156.32 176.24 ΞDG 1207.33 169.07 835.23 1331.34 158.64 173.47 minf2 ΞDG 1144.36 179.03 744.71 1144.31 163.82 179.03 minf1 DG 0 Ξel bow 1197.79 176.62 749.21 1197.74 160.93 176.62 ΞDG 1307.33 172.87 828.55 1307.35 159.78 172.87 minf2

Figure 12 shows a bubble plot representation of the selected optimal solutions. The axes report the EENS and ECg values while the diameters of the bubbles are proportional to their respective CVaR values. The MS case is also plotted.

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From Table 6 and Figure 12 it can be seen that, the MS case has an expected performance

(EENS = 1109.21 (kW h) and ECg = 170.27 ($)) inferior (high EENS and ECg ) to any case for which DG is optimally integrated. Furthermore, the CVaR(ENS) = 1656.53 (kW h) for the MS case is the highest, indicating the high risk of actually achieving the expected performance of energy not supplied. This conﬁrms that DG is capable of providing a gain of reliability of power supply and economic beneﬁts, the risk of falling in scenarios of large amounts of energy not supplied being reduced.

175 AB175

Ø ∝ CVaRα(ENS) Ø ∝ CVaRα(Cg)

165 165 betaβ = 1 1 bβeta= 0.75 075 ($) [$]

g g betaβ = 0.5 05 EC EC bβeta= 0.25 02 5 155 155 betaβ = 0 0 msMS

145 145 600 800 1000 1200 600 800 1000 1200 EENS (kWh) EENS (kWh)

Figure 12: Bubble plots EENS v/s ECg . Diameter of bubbles proportional to CVaR(ENS) (A) and CVaR(C g) (B)

Comparing among the selected optimal DG–integrated networks, in general the expected performances of EENS and ECg are progressively lower for increasing β. This to be expected: lowering the values of β, the MOO tends to search for optimal allocations and sizing ΞDG that sacriﬁce expected performance at the beneﬁt of decreasing the level of risk (CVaR). These insights can serve the decision making process on the integration of renewable DG into the network, looking not only at the give–and–take between the values of EENS and, but also at the level of risk of not achieving such expected performances due to the high variability.

Figure 13 shows the average total DG power allocated in the distribution network and its breakdown by type of DG technology for the optimal ΞDG as a function of β. It can be pointed out that the contribution of EV is practically negligible if compared with the other technologies. This is due to the fact that the probability that the EV is in a discharging state is much lower than that of being in the other two possible operating states, charging and disconnected (see Figure 10), combined with the fact that when EV is charging the effects are opposite to those desired.

27 129 needs to be installed. However, focusing on the individual fractions of average power allocated by PV, W and ST (Figure 13(B), (C) and (E), respectively), show that a reduction of the risk in the

EENS and ECg is achieved speciﬁcally diminishing the proportion of PV power (from 0.29β=1 to

β B C

P pv Pw

PT PT

β β D E

Pev Pst

PT PT

β β

Figure 13: Average total DG power allocated (A) and its breakdown by type of DG: PV (B), W (C), EV (D) and ST (E)

Table 7 summarizes the minimum, average and maximum total renewable DG power allocated per node. The tendency is to install more localized sources (mainly nodes 4 and 8) of renewable DG power when the MOO searches only for the optimal expected performances (β = 1) and to have a more uniformly allocation of the power when searches for minimizing merely the CVaR (β = 0).

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Table 7: Average, minimum and maximum total DG power allocated per node

β PT (kW) 1 0.75 0.5 0.25 0 node min mean max min mean max min mean max min mean max min mean max 1 12.08 34.44 54.77 1.15 22.40 38.56 0.00 19.23 40.98 0.00 39.03 121.00 3.00 17.33 34.71 2 2.30 40.72 69.73 0.00 49.95 77.70 36.50 58.40 123.36 3.00 63.61 132.93 0.00 42.54 84.09 3 0.00 24.83 46.45 14.80 41.79 85.03 0.00 37.94 105.11 4.00 36.87 98.53 1.00 32.84 77.78 4 76.00 110.00 133.41 1.15 67.40 133.63 0.58 38.04 80.13 6.15 20.73 61.85 0.00 39.85 85.86 5 22.60 52.39 77.08 28.90 60.66 98.59 12.63 89.39 143.50 3.30 23.49 54.25 1.00 24.97 79.64 6 12.33 55.56 85.46 10.45 21.22 38.95 2.00 27.68 106.26 12.15 53.78 84.43 0.00 50.64 116.85 7 8.00 16.52 35.38 39.38 64.07 104.05 0.00 52.03 159.73 0.00 34.09 92.81 5.00 18.51 39.23 8 79.03 111.20 146.63 30.00 74.57 114.41 0.00 40.60 146.06 4.00 37.94 102.60 1.00 39.49 119.38 9 0.00 20.03 68.73 4.00 74.07 107.88 0.00 46.72 85.61 0.00 44.06 94.08 0.00 32.86 74.53 10 0.00 9.07 25.35 0.00 1.58 7.88 0.00 11.88 58.69 0.00 8.58 43.40 0.00 30.12 83.45 11 0.00 9.98 17.68 0.00 3.04 13.20 0.00 4.74 23.45 0.00 8.99 45.95 0.00 7.31 51.17

5 Conclusions

We have presented a risk-based simulation and multi-objective optimization framework for the integration of renewable generation into a distribution network. The inherent uncertain behavior of renewable energy sources and variability in the loads are taken into account, as well as the possibility of failures of network components. For managing the risk of not achieving expected performances due to the multiple sources of uncertainty, the conditional value-at-risk is introduced in the objective functions, weighed by a β parameter which allows trading off the level of risk. The proposed framework integrates the Non-dominated Sorting Genetic Algorithm II as a search engine, Monte Carlo simulation to randomly generate realizations of the uncertain operational scenarios and Optimal Power Flow to model the electrical distribution network ﬂows. The optimization is done to simultaneously minimize the energy not supplied and global cost, combined with their respective conditional value-at-risk values in an amount controlled by β.

To exemplify the proposed framework, a case study has been analyzed derived from the IEEE 13 nodes test feeder. The results obtained show the capability of the framework to identify Pareto optimal sets of renewable DG units allocations. Integrating the conditional value-at-risk into the framework and performing optimizations for different values of β has shown the possibility of optimizing expected performances while controlling the uncertainty in its achievement. The contribution of each type of renewable DG technology can also be analyzed, indicating which is more suitable for speciﬁc preferences of the decision makers.

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Paper (ii)

R. Mena, M. Hennebel, Y.–F. Li and E. Zio, “Self–adaptable hierarchical clustering analysis and differential evolution for optimal integration of renewable distributed generation,” Applied Energy, vol. 133, pp. 388—402, 2014. 138

Self-adaptable hierarchical clustering analysis and differential evolution for optimal integration of renewable distributed generation

Rodrigo Menaa, Martin Hennebelb, Yan-Fu Lia, Enrico Zioa,c,1

aChair on Systems Science and the Energetic Challenge, Fondation Electricité de France at CentraleSupeléc, Chatenay-Malabryˆ Cedex, France bCentraleSupeléc, Department of Power & Energy Systems, Gif-Sur-Yvette, France cPolitecnico di Milano, Energy Department, Milan, Italy

Abstract

In a previous paper, we have introduced a simulation and optimization framework for the integration of renewable generators into an electrical distribution network. The framework searches for the optimal size and location of the distributed renewable generation units (DG). Uncertainties in renewable resources availability, components failure and repair events, loads and grid power supply are incorporated. A Monte Carlo simulation and optimal power flow (MCS–OPF) computational model is used to generate scenarios of the uncertain variables and evaluate the network electric performance with respect to the expected value of the global cost (ECG). The framework is quite general and complete, but at the expenses of large computational times for the analysis of real systems. In this respect, the work of the present paper addresses the issue and introduces a purposely tailored, original technique for reducing the computational efforts of the analysis. The originality of the proposed approach lies in the development of a new search engine for performing the minimization of the ECG, which embeds hierarchical clustering analysis (HCA) within a differential evolution (DE) search scheme to identify groups of similar individuals in the DE population and, then, ECG is calculated for selected representative individuals of the groups only, thus reducing the number of objective function evaluations. For exemplification, the framework is applied to a distribution network derived from the IEEE 13 nodes test feeder. The results show that the newly proposed hierarchical clustering differential evolution (HCDE) MCS–OPF framework is effective in finding optimal DG–integrated network configurations with reduced computational efforts.

Keywords: Renewable distributed generation, uncertainty, simulation, optimization, differential evolution, hierarchical clustering analysis

1 Corresponding author. Tel: +33 1 4113 1606; fax: +33 1 4113 1272. E-mail addresses: [email protected], [email protected] (E. Zio).

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1 Introduction

Renewable distributed generation (DG) requires the selection of the different available technologies, and their sizing and allocation onto the power distribution network, considering the specific economic, operational and technical constraints [1–5]. This can become a complex optimization problem, depending on the size of the distribution network and the number of renewable DG technologies available, that can lead to combinatorial explosion [1,3, 6–9]. Furthermore, for each renewable DG plan considered, the power flow problem needs to be solved to assess the response of the distribution network in terms of power and voltage profiles, available power usage, power demand satisfaction, economic performances, etc., with possibly significant computation times.

Heuristic optimization techniques belonging to the class of Evolutionary Algorithms (EAs), like honey bee mating [10], particle swarm optimization (PSO) [9,11–13], differential evolution (DE) [14, 15] and genetic algorithms (GA) [2, 3, 16, 17], have been considered for the solution to this problem, since they can deal straightforwardly with non–convex combinatorial problems, discontinuous search spaces and non–differentiable objective functions [1, 9]. To improve the performance of EAs for the complex optimization problem of DG planning, we consider the integration of clustering [18–23]. This can be directed to the enhancement of the global and/or local searching ability of the algorithm, and amounts to identifying groups of similar individuals and applying different evolution operators to those of a same cluster (group) [18,20–22], e.g. for random generation of new individuals in the neighborhood of cluster centroids [23], or multi–parents crossover over new randomly generated individuals spread in the global feasible space [19]. Even if convergence is improved, some of these methodologies increase temporarily the overall size of the population and, thus, the computational effort. In addition, the accuracy of the clusters structures in representing the distribution of individuals must be controlled for performing clustering conveniently.

The main original contribution of the work here presented, lies in the development of the clustering strategy in a controlled manner. The implementation of such clustering strategy is done within a Monte Carlo simulation and optimal power ﬂow (MCS–OPF) model and differential evolution (DE) optimization framework [24] previously developed by the authors for the integration of renewable generators into an electrical distribution network: the framework searches for the optimal size and location of the distributed renewable generation units (DG) [25]. Optimality of the DG plan is sought with respect to the expected global cost (ECG). The introduction of the clustering is hierarchically (i.e., hierarchical clustering analysis, HCA, [26]) by a controlled way of reducing the number of individuals to be evaluated during the DE search, therefore, improving the computational efﬁciency. Henceforth, we call our method hierarchical clustering differential evolution (HCDE).

HCA is introduced to build a hierarchical structure of grouping individuals of the population

2 140 that present closeness under the control of a specific linkage criterion based on defined distance metrics [26]. The HCA outcomes are the linkage distances at which the grouping actions take place, defining the different levels in the hierarchical structure. Two control parameters are introduced in the HCA, the cophenetic correlation coefficient (CCC) and a cutoff level coefficient of the linkage distances in the hierarchical structure of the groups (pco). The CCC is a similarity coefficient that measures how representative is the proposed grouping structure by comparing their linkage distances with the original distances between all the individuals in the population. In the hierarchical structure, the linkage distance given by pco sets the level at which the groups formed below it are considered to be ‘close enough’ to constitute independent clusters. The two parameters allow HCDE to adapt itself in each generation of the search, ‘deciding’ whether to perform clustering if the CCC is greater than or equal to a preset threshold (CCCth) and cutting the hierarchical structure in independent clusters according to the linkage distance given by pco. Then, the individual closest to the centroid of each cluster is taken as the feasible representative solution in the population that enters the evolution phase of the HCDE algorithm. Figure 1 summarizes schematically the structure of the proposed framework.

Random generation of DE initial population

Objective function ECG evaluation for each individual in the population through MCS−OPF

Construction of a hierarchical structure of groups of individuals in the population by performing HCA and calculation of the corresponding CCC

no Is CCCth ≤ CCC? yes

Cutting off the hierarchical structure according to pco forming clusters of individuals, identifying the individual closest-to-the-centroid for each cluster to form a reduced population to enter the DE evolution phase

Retention of the complete population

Generation of an evolved population producing trial individuals through DE mutation and cross over operators, evaluating the objective function by performing MCS−OPF and retaining the or replacing for the best individuals

Is the DE stopping no criteria fulfilled? yes The best individual in the population is returned

Figure 1: HCDE framework schema

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We test the approach on a case study based on the IEEE 13 nodes test feeder distribution network [27], completing the study with a sensitivity analysis to investigate the effects of the parameters controlling the clustering, namely CCC and pco. For practical ease of the presentation of the approach, in the next section we provide the basic elements of the model of the distribution network considered as case study and we brieﬂy summarize the MCS–OPF model taken from [25]. In Section 3, we embed this in the HCDE for renewable DG selection, sizing and allocation. Finally, in Section 4 we present the numerical results of the case study and in Section 5 we draw some conclusions on the work performed.

2 Renewable DG–integrated network model

The operation of the renewable DG–integrated network is considered to be dictated by the location and magnitude of the power available in the different sources, the loads and the operating states of the components. Uncertainty is present in the states of operation of the components, due to stochasticity of degradation and failures, and in the behavior of the renewable energy sources. These uncertainties have a direct impact on the power available (from the DG units, main supply spots and/or feeders) to satisfy power demands, which are, in turn, also subject to ﬂuctuations. Furthermore, if the distribution network is considered as a ‘price taker’ entity, the uncertain behavior of the power demand impacts directly over the energy price [4,5, 28]. Consequently, an attentive modeling of the uncertainties in renewable DG planning is imperative for well–supported decision– making.

Monte Carlo simulation (MCS) has already been used to emulate the stochastic operating conditions and evaluate the performance of power distribution networks [19,28–30]. In the present paper, non–sequential MCS is used to randomly sample the modeled uncertain variables for a speciﬁc renewable DG plan, without dependence on previous operating conditions, characterizing the network operation in terms of location and magnitudes of power available and loads. Then, the performance of the DG–integrated network is evaluated through the optimal power ﬂow model.

2.1 Monte Carlo and optimal power ﬂow simulation

In the proposed framework, the renewable DG technologies considered are of four types: solar photovoltaic (PV), wind turbines (W), electric vehicles (EV) and storage devices (ST); these are represented by the set DG that contains all the d g types of technologies. As for main power supply spots or transformers (MS), the set MS indicates the ms different types of MS considered in the network.

The DG–integrated network deployment is represented by the location and capacity size of the power sources, as indicated in matrix form in Equation (1) below, where ξi,j indicates the number

4 142 of units of main supply spots or DG technology j that are allocated at a node i:

ξ1,1 ξ1,j ξ1,ms ξ1,1+ms ξ1,j+ms ξ1,d g+ms . ···. . ···. . . ···. . ···......   MS DG Ξ = ξi,1 ξi,j ξi,ms ξi,1+ms ξi,j+ms ξi,d g+ms = [Ξ Ξ ]  . ···. . ···. . . ···. . ···. .  | (1)  ......    ξ ξ ξ ξ ξ ξ   n,1 n,j n,ms n,1+ms n,j+ms n,d g+ms   ··· ··· ··· ···  ξi, j Z∗, i N, j PS ∀ ∈ ∈ ∈ where N and PS = MS DG are the set of nodes in the network and the set of all power sources, ∪ whose cardinalities are n and ps = ms + d g, respectively.

The set of feeders FD is deﬁned by all the pairs of nodes (i, i') connected by a distribution line (i, i') N N. ∀ ∈ × The considered uncertain conditions that determine the operation of the DG–integrated network are accounted for using different stochastic models, as summarized in Table 1. The interested reader can consult [25] for further details.

Table 1: Uncertain conditions models in the DG–integrated network operation

Variable Nomenclature States and units Model Parameters

Hour of the day td (h) Discrete uniform distribution [1, 24] F R (0): under repair λj , λj Mechanical state mci,j Two–state Markov F R (1): operating λi,i', λi,i' Truncated normal distribution µMS , σMS Main power supply P MS (kW) j j i,j 0 P MS P MS P MS i,j capj capj ≤ ≤ PV PV Solar irradiance si [0, 1] Beta distribution αi , βi W Wind speed wsi (m/s) Rayleigh distribution σi ‘Block groups’ Hourly probability distribution (-1) : charging of EV operating states per day: EV operating state opEV (0): disconnected t i,j p (charging) d (1): discharging − p0 (disconnected) p+ (discharging) ST level of charge QST (kJ) Uniform distribution SEST M ST i,j [ j Ti,j ] Daily nodal load proﬁles,

hourly normally distributed load. L L Nodal power demand Li (kW) µ (td ), σ (td ) Truncated normal distribution i i

0 Li ≤ ≤ ∞ F R where i, i' N, j PS, (i, i') FD, λj and λj (1/h) are the failure and repair rates of the power source j, respectively, ∀ ∈ ∈ ∈ F R MS MS λi,i' and λi,i' (1/h) are the failure and repair rates of the feeder (i, i'), respectively, µj and σj (kW) are the normal distribution mean and standard deviation associated to the main supply j, P MS (kW) is the maximum capacity of the capj PV PV transformer j, αi and βi are the parameters of the Beta probability density function of the solar irradiance at node i, W ST σi is the scale parameter of the Rayleigh distribution function of the wind speed at node i, SEj (kJ/kg) is the speciﬁc

5 143 energy of the active chemical in the battery type j, M ST (kg) is the mass of active chemical in the battery type j at node i, Ti,j L L µi (td ) and σi (td ) (kW) are the hourly mean and standard deviation of the normal distribution of the power load at node i.

Concerning the hour of the day td (h), sampled from a discrete uniform distribution U(1, 24), the night interval is deﬁned between 22.00 and 06.00 h. If the value of td falls in the night interval, there is no solar irradiation.

The resulting realization of one operational scenario of duration ts (h), for the given DG plan denoted by FD, Ξ , consists in the random sampling of each uncertain variable (Table 1), here indicated by{ the vector} ϑ below:

MS EV ST ϑ = [td , mci,j, mci,i', Li, Pi,j , si, wsi, opi,j ,Qi,j ] (2)

To evaluate the performance of the distribution network the OPF model receives as input the location and magnitude of the available power in the power sources and demanded at the loads, which are set by the operating conditions defined by FD, Ξ and ϑ. The nodal power loads Li are directly sampled, whereas the available power in the{ power} sources (MS and DG) depends on the uncertain variables that represent the behavior of the energy sources, the specific technical characteristics of each type of technology and the mechanical states. The available power in each type of power source considered is modeled by the functions summarized in Table 2, for a given configuration FD, Ξ , operating scenario ϑ and a generic node i. { } Table 2: Available power functions of the power sources (PS) [25, 29, 30]

PS type j Parameters Available power function (kW) MS;ϑ ϑ MS;ϑ MS - Pai,j = ξi,j mci,j Pi,j (3) T PV ;ϑ ϑ ϑ ϑ ϑ 3 ai Pai,j (si ) = ξi,j mci,j FFj Vi,j Ii,j 10− (4) × NoTj ϑ ϑ T = Tai + s (NoT 20)/0.8 I ci,j i j PV scj − V I ϑ sϑ I k T ϑ 25 ocj i,j = i ( scj + I j ( ci,j )) k , k ϑ ϑ− I j Vj V V k T i,j = ocj + Vj ci,j VMPPj , IMPPj FFj = (VMPPj IMPPj )/(Vocj Iscj )

wsϑ ws ws i cij cij PW − if ws wsϑ < ws R j wsa wsci cij i aj wsa j j W j W;ϑ ϑ ϑ − ≤ Pa ws ξ mc  W ϑ (5) wsco i,j ( i ) = i,j i,j P if wsa ws wsco j ×  R j j i j PW  ≤ ≤ R j 0 otherwise

EV ;ϑ EV ;ϑ  ϑ EV ;ϑ EV EV  EV ;ϑ EV t EV ;ϑ , P Pai,j (opi,j , t) =ξi,j mci,j opi,j PR t [0, t ] (6) op R j opi,j i,j j ∀ ∈ PaST;ϑ t ξ mcϑ PST t 0, tST;ϑ (7) i,j ( ) = i,j i,j R j [ Ri,j ] ST PST ∀ ∈ R j tST;ϑ QST;ϑ QST;ϑ/PST Ri,j ( i,j ) = i,j R j

MS;ϑ ϑ In Table 2, Pai,j (kW), ξi,j and mci,j denote the available power, the units and the mechanical

6 144 state of the power source of type j allocated at node i. For solar photovoltaic technologies j PV , the parameter T ( C) is the ambient temperature at node i, N ( C) is the nominal cell operation∈ ai ◦ oTj ◦ temperature, I (A) is the short circuit current, V (V) is the open circuit voltage, k (mV/ C) is scj ocj Vj ◦ the voltage temperature coefﬁcient, k (mA/ C) is the current temperature coefﬁcients and V I j ◦ MPPj (V) and I (A) are the voltage and current at maximum power point, respectively. For wind MPPj turbines of types j W, ws , ws and ws (m/s) are the cut–in, rated and cut–out wind speeds, cij aj coj W∈ respectively, and PR (kW) is the rated power of the turbine. For electric vehicles j EV , topEV ;ϑ (h) j ∈ i,j is the time of residence in the operating state opEV ;ϑ and P EV (kW) is the rated power. For storage i,j R j devices j ST, tST;ϑ (h) is the upper bound of the discharging time interval and PST (kW) is the Ri,j R j rated power.∈

Under the operating conditions set forth, the given configuration of the renewable DG–integrated network FD, Ξ and the scenario ϑ, the OPF objective is the minimization of the operating cost associated{ to the} generation and distribution of power, considering the revenues per kW h sold. Power flow analysis is performed by DC modeling, neglecting power losses and assuming the voltage throughout the network as constant, linearizing the classic non–linear power flow formulation by accounting solely for active power flows [31,32]. The present formulation of the DC optimal power flow problem is:

ϑ PS ϑ min Co (Pu, ∆δ) = (Covj ep )Pui,j i N j PS − X∈ X∈ (8) FD ϑ + Covi,i' Bi,i'(δi δi') + (Cop + ep ) LSi (i,i') FD | − | i N X∈ X∈ s.t.

ϑ ϑ Li LSi Pui,j mci,i'Bi,i'(δi δi') = 0 (9) − − j PS − i N − X∈ X∈ PS;ϑ 0 Pui,j Pai,j (10) ≤ ≤ NET FD Bi,i'(δi δi') V Ai,i' (11) | − | ≤ ϑ where i, i' N, j PS, (i, i') FD and the operating scenario ϑ, Co ($/h) is the operating cost of ∀ ∈ ∈ ∈ PS the total power supply and distribution, Covj ($/kWh) is the variable operating cost of the power ϑ source j, ep ($/kWh) is the energy price, Pui,j (kW) is the used power from the source of type j at FD 1 node i, Covi,i' ($/kWh) and Bi,i' (Ω− ) are the variable operating cost and the susceptance of the feeder (i, i'), respectively, δi is the voltage angle at node i, Cop ($/kWh) is the opportunity cost for NET kW h not supplied, V (kV) is the nominal voltage of the network and Ai,i' (A) is the ampacity of the feeder (i, i'). The load shedding LSi (kW) is deﬁned as the amount of load disconnected at node i to alleviate congestions in the feeders and/or balance the demand of power with the available power supply.

7 145

The distribution network is considered as a ‘price taker’ entity, assuming a correlation between the total demand of power and the energy price ep ($/kWh). Then, the energy price is calculated from an intermediate correlation proposed by [4, 5, 28]:

2 TL(td ) TL(td ) ep(TL) = eph 0.38 + 1.38 (12) − TLh TLh where eph is the energy price corresponding to the highest value of total demand considered TLh. The total demand of power TL(td ) at the hour of the day td is the summation of all the nodal loads Li(td ) (Table 1). The constraint given by Equation (9) corresponds to the power balance equation at node i, whereas Equations (10) and (11) represent the bounds of the power generation and technical limits of the feeders, respectively.

One realization of the MCS–OPF consists of the sampling of NS operating scenarios ϑ regarded as the set Υ = ϑ1,..., ϑh,..., ϑNS for each of which the optimal power ﬂow problem is solved, giving in output{ the values of the minimum} operating cost of the total power supply and distribution Υ ϑ ϑ ϑ Co = Co 1 ,..., Co h ,..., Co NS . { }

2.2 Expected global cost ECG

The proposed renewable DG–integrated network solutions are evaluated with respect to the expected global cost ECG. The global cost CG is composed by two terms: the ﬁxed investment and operation (maintenance) costs Ci ($), which are prorated hourly over the life of the project th (h), and the operating costs CoΥ ($/h) that is the outcome of the MCS–OPF (Equation (8)) described in the precedent Section 2.1. Thus, the global cost function for a scenario ϑ is given by:

ϑ ϑ CG = Ci + Co ϑ Υ (13) ∀ ∈ 1 Ci ξ ci (14) = th i,j j i N j DG X∈ X∈ where cij ($) is the investment cost of the DG technology type j.

Υ ϑ ϑ ϑ Then, the global cost CG = CG 1 ,..., CG h ,..., CG NS is considered as realizations of the probability mass function of CG,{ and from multiple realizations} the expected value ECGΥ can be obtained.

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3 Renewable DG selection, sizing and allocation

The aim of the proposed simulation and optimization framework is to find the optimal plan of integration of renewable DG in terms of selection, sizing and allocation of generation units from different technologies available (PV, W, EV and ST). The corresponding decision variables are contained in ΞDG of the configuration matrix Ξ defined in Equation (1).

3.1 Optimization problem formulation

Considering a network conﬁguration (FD, Ξ) and a set of randomly generated scenarios Υ , the optimization problem is formulated as follows:

min ECGΥ (15)

s.t.

ξi,j Z∗ (1) ∈ ξi,j cij BGT (16) i N j DG ≤ X∈ X∈ ξi,j τj (17) i N ≤ X∈ MCS OPF((FD, Ξ), Υ ) (18) −

The meaning of each constraint i, i' N, j PS, (i, i') FD, τj Z∗ is: ∀ ∈ ∈ ∈ ∈

• (1): the decision variable ξi,j is a non–negative integer number. • (16): the total investment and ﬁxed operation and maintenance costs must be less than or equal to the available budget BGT.

• (17): the total number of renewable DG units of each technology j to be allocated must be

less than or equal to the maximum number of units available for integration τj. • (18): all Equations (8)–(11) of MCS–OPF must be satisﬁed.

3.2 Hierarchical clustering differential evolution (HCDE)

The complex combinatorial optimization problem of DG planning under uncertainties described above is solved by integrating DE with HCA to reduce computational efforts, whereby the evaluation of the objective function is performed by the MCS–OPF presented in Section 2.1.

DE is a population–based and parallel, direct search method, shown to be one of the most efﬁcient

9 147 evolutionary algorithms to solve complex optimization problems [19, 21, 24]. The implementation of the original version of DE involves two main phases: initialization and evolution, summarized below for completeness of the paper [24]:

Initialization

Set the values of parameters: ◦ • NP: population size

• Gmax : maximum number of generations • Coc: crossover coefﬁcient [0, 1] ∈ • F: differential variation ampliﬁcation factor [0, 2] ∈ Generate randomly NP individuals X (decision vectors) within the feasible space, to ◦ 0 0 0 0 form the initial population POP = X1 ,..., Xk ,..., XNP { } Evaluate the objective function f (X ) = y for each individual ◦ Evolution loop

Set generations count index G = 1 ◦ G 0 Set POP = POP ◦ While G Gmax (stopping criterion) ◦ ≤ Trial loop G G For each individual Xk in POP , k 1, . . . , NP : ∀ ∈ { } • Sample from the uniform distribution three integer indexes r1, r2, r3, with k = r1 = r r and choose the corresponding three individuals X G , X G , X G 6 6 2 = 3 r1 r2 r3 6 G • Mutation: Generate a mutant individual Vk according to:

G G G G Vk = X r + F(X r X r ) (19) 1 2 − 3 G • Crossover: initialize a randomly generated vector Uk , whose dimensionality dim G G is the same as that of Xk and each coordinate uk,i follows a uniform distribution with outcome in [0, 1] i 1,..., dim . In addition, generate randomly an integer index ri 1,..., dim∀ from∈ { a uniform} distribution to ensure that at least one ∈ { G } G coordinate from Vk is exchanged to form a trial individual XTk , whose coordinates are deﬁned as follows:

G G G vk,i if uk,i Cco or i = ri x t = ≤ (20) k,i  G G xk,i if uk,i > Cco and i = ri  6 • Selection: evaluate the objective function for the trial individual f XT G ; if f XT G  ( k ) ( k ) G G G G < f (Xk ) (minimization), then XTk replaces Xk in the population POP , otherwise G Xk is retained

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Set G = G + 1 ◦ Once the stopping criterion is reached, sort the individuals in POPGmax in descending ◦ order according to their values of the objective function and return X Gmax

The original version of DE keeps the population size NP constant, making the computational performance dependent mainly on the number of objective function evaluations carried out during the evolution phase of the algorithm. Then, the integration of HCA into DE is aimed at the reduction of the number of individuals that enter the evolution loop in each generation so as to decrease the number of objective function evaluations.

HCA links individuals or groups of individuals which are similar with respect to a speciﬁc property, translated into a metric of distance, obtaining a hierarchical structure. In practice, we use an agglomerative procedure which in sp = NP 1 steps fuses the closest pair or individuals or groups of individuals through a linkage function, e.g.− single linkage (nearest neighbor distance), complete linkage (furthest neighbor), average linkage, among others, until the complete hierarchical structure is built. The base hierarchical clustering algorithm used in this study can be expressed as follows [26]:

Step 0: Given a population POP POP = X1,..., Xk,..., XNP , form the set of singleton groups { } O = Op = Xk , p = k 1,..., NP and calculate the linkage distances between all the NP groups{ { using}} the∀ average∈ { as linkage} function and the Euclidean distance as metric:

1 1 1 0 d1,2 d1,q d1,NP . ···. . ···......   X X 2 0 d1 d1 ( kp kq) 1 p,q p,NP 1 Xkp Op Xkq Oq − ∈ ∈ D =  ···. .  dp,q = P P Æ  0 .. .  Op Oq (21)    .  | || |  .. d1   NP 1,NP   −0      p, q 1, . . . , NP , kp, kq 1, . . . , NP ∀ ∈ { } ∈ { } 1 where dp,q is the average of the Euclidean distances between all the individuals Xk belonging to the groups Op and Oq, respectively. 1 Step 1: Fuse the ﬁrst pair of groups Op' and O1', for which dp',q' is the minimum distance 1 min(D ) and form a new group ONP+1 = Op' Oq' . Update the set of groups O replacing Op' { ∪ } 2 and Oq' by ONP+1, and calculate the linkage distances D between all the NP 1 groups in O using (21). −

2 Step 2: Fuse the second pair of groups Op' and Oq' for which dp',q' is the minimum distance 2 min(D ), and form a new group ONP+2 = Op' Oq' . As in the preceding step, update the set of groups O and calculate the linkage distances{ ∪D3 }between all the NP 2 groups in O using (21). −

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. .

NP 1 Step NP 1: Fuse the last pair of groups with linkage distance dp',q'− , forming the last group − O2NP 1 = Op' Oq' that contains all the individuals X . − { ∪ } The outcoming hierarchical (or tree) structure can be reported as a sorted table containing the NP 1 linkage distances relative to each pairing action of individuals/groups and be graphically illustrated− as a dendrogram. Table 3 and Figure 2 present, respectively, the resultant linkage distances and dendrogram obtained from an example set of NP = 8 two–dimensional individuals X using the above introduced HCA algorithm.

Table 3: Example hierarchical structure outcome

Step sp Group Groups linked Linkage distance

1 1 O9 O2 O6 = X2 X6 d2,6 { ∪ } {{ } ∪ { }} 2 2 O10 O3 O4 = X3 X4 d3,4 { ∪ } {{ } ∪ { }} 3 3 O11 O1 O7 = X1 X7 d1,7 { ∪ } {{ } ∪ { }} 4 4 O12 O5 O8 = X5 X8 d5,8 { ∪ } {{ } ∪ { }} 5 5 O13 O9 O11 = X2, X6 X1, X7 d9,11 { ∪ } {{ } ∪ { }} 6 6 O14 O10 O12 = X3, X4 X5, X8 d10,12 { ∪ } {{ } ∪ { }} 7 7 O15 O13 O14 = X1, X2, X6, X7 X3, X4, X5, X8 d13,14 { ∪ } {{ } ∪ { }}

7 5 d13,14 3.0 O15 X8 4 X5 X 3 2.5 6 d10,12 X4 O14 3 2.0 xj d 2 1.5 5 X2 d9,11 O 13 4 X6 d X7 3 5,8 2 O11 O 1 1.0 d1,7 d 12 3,4 O X1 1 10 d2,6

0 0.5 O9 0 1 2 3 4 5 O2 O6 O1 O7 O3 O4 O5 O8 xi

Figure 2: Example dendrogram for average linkage HCA

As stated above, HCA builds the hierarchical structure through a linkage function introducing in each grouping action a larger or smaller degree of distortion with respect to the original distances between (ungrouped) individuals. The measurement of this distortion is important and the cophenetic correlation coefﬁcient (CCC)is introduced to evaluate how representative is the

12 150 hierarchical structure proposed by the HCA. The CCC can be obtained from Equations (22) and (23) below [26].

1 ¯1 ¯ (dp,q D )(hp,q H) p

Whether the hierarchical structure is accepted, the clustering process itself takes place. As sp before stated, the HCA outcome linkage distances dp',q' define each level (height) at which a pairing action takes place. If the hierarchical structure is ‘cut off’ at a specific linkage distance dCO, all the groups that are formed below the level dCO become independent clusters. In each generation G G of HCDE, a dCO relative to the HCA outcome linkage distances for the corresponding POP , is determined from a preset cutoff level coefficient pco of the linkage distances between the minimum sp dp',q' that correspond to the first pairing action and the distance to form at least four clusters needed to perform the mutation process in the HCDE. Thus, dCO can be obtained from Equations (24) and (25). Figure 3 shows the cutoff distance representation for the example aforementioned, for which

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the formed clusters are O2, O6 , O1 , O7 , O3, O4 , O5 and O8 . { } { } { } { } { } { }

dCO = dmin + pco(dNC=4 dmin) (24) − dNC 4 = d 1 4 %ile (25) = ( NP ) −

3.0

2.5

2.0 d 1.5

d NC =4 1.0 dCO dmin 0.5 O2 O6 O1 O7 O3 O4 O5 O8

Figure 3: Example of cutoff distance calculation

The integration of HCA into DE and the deﬁnition of the parameters CCCth and pco allow HCDE adaptation at each generation, i.e., deciding whether to perform HCA and determining the clusters to be taken. Then, the individuals closest to the centroids of the formed clusters are considered as the representatives of the group which they belong to and are taken in a reduced population that enters the evolution phase of the HCDE. The proposed HCDE algorithm is summarized schematically in the ﬂowchart of Figure 4.

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Set the values of NP, Gmax, Coc, F, CCCth and pco

Generate randomly NP individuals ΞDG, according to constraints (1), (16) 0000DG; DG; DG; and (17), to form the initial population POP {ΞΞΞ1 , ,kNP , , }

Evaluate the objective function ECGϒ for each individual through MCS-OPF((FD,Ξ), ϒ) with Ξ  [ ΞΞ M SDG ]

Set generations count index G = 1

sp Perform HCA using the average distance linkage function obtaining d p',q' for the NP-1 pairing actions sp and calculate the cophenetic correlation index CCCG

no G Is CCCth ≤ CCC ? yes

Cut off the hierarchical structure according to pco ((24) and (25)) forming NPG clusters. Obtain the individual closest-to-the-centroid for each group ck and set POP'GDG;GDG;GDG;G={ΞΞΞ ,, ,, } 1 ck NPG

G G G DG;G DG;G DG;G Set NP = NP and POP'  POP{ΞΞΞ1 , ,kNP , , }

DG;G G G Perform the trial loop, from eachΞk' POP' , k'{1 ,...,NP } generate a G trial individual ΞΤk' applying mutation and cross over operators ((19) and (20),  G respectively) and evaluate the objective function ECG ()ΞΤk'

no GG Is ECG(ΞΤk')( ECG Ξ k' ) ? yes

G G G GG Replace Ξ k by ΞΤ k' in POP where k is such that ΞΞkk'

GG G ΞΞkk' is retained in POP

Set G = G+1

no Is G ≤ Gmax? yes Sort the individuals in POP G in descending order according to their

Gmax values of objective function and return Ξ1

Figure 4: Flowchart of the framework

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4 Case study

We consider a modiﬁcation of the IEEE 13 nodes test feeder distribution network [27] with the original spatial structure but neglecting the feeders of length zero, the regulator, capacitor and switch. The resulting network has 11 nodes and presents the relevant characteristics of interest for the analysis, e.g. the presence of a main power supply spot and comparatively low and high spot, and distributed load values [33].

4.1 Distribution network description

The distribution network presents a radial structure of n = 11 nodes as shown in Figure 5. The nominal voltage V NET is 4.16 (kV), kept constant for the resolution of the DC optimal power ﬂow problem.

MS i = 1

54 2 633-6343

98 671-6926 7

10 11

Figure 5: Radial 11–nodes distribution network

Table 4 contains the technical characteristics of the different types of feeders considered: specif- FD ically, the indexes of the pairs of nodes (i, i') that they connect, their length l, reactance X , ampacity AFD and failure and repair rates.

The nodal power demands are built from the load data given in [27] and reported in Figure 6 as L L daily proﬁles, normally distributed on each hour td with mean µ and standard deviation σ [29,34]. The technical parameters, failure and repair rates and costs of the MS and the four different types of DG technologies (PV,W, EV and ST) available to be integrated into the distribution network are given in Table 5. For the present case study, the distribution region is such that the solar irradiation and wind speed conditions are assumed uniform in the whole network, i.e., the values of the parameters of the corresponding Beta and Rayleigh distributions are assumed constant in the whole network.

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Table 4: Feeders characteristic and technical data [11, 27, 28]

Type Node i Node i' l (km) X FD (Ω/km) AFD (A) λF (1/h) λR (1/h) CovFD ($/kWh) T1 1 2 0.610 0.371 730 3.333e 04 0.198 1.970e 02 T2 2 3 0.152 0.472 340 4.050e−04 0.162 9.173e−03 T3 2 4 0.152 0.555 230 3.552e−04 0.185 6.205e−03 T1 2 6 0.610 0.371 730 3.333e−04 0.198 6.205e−03 T3 4 5 0.091 0.555 230 3.552e−04 0.185 6.205e−03 T6 6 7 0.152 0.252 329 4.048e−04 0.164 8.904e−03 T4 6 8 0.091 0.555 230 3.552e−04 0.185 1.970e−02 T1 6 11 0.305 0.371 730 3.333e−04 0.198 1.970e−02 T5 8 9 0.091 0.555 230 3.552e−04 0.185 9.173e−03 T7 8 10 0.244 0.318 175 3.552e−04 0.185 6.205e−03 − − 1800 180

node 2 1350 135node 2 node 3 node 3 node 4 node 4 900 (kW) 90 node 5 (kW) L L node 5 μ σ node 6 node 6 node 7 450 45 node 7 node 9 node 9 node 10 0 node0 10 0 4 8 12162024 0 4 8 12 16 20 24

td (h) td (h)

Figure 6: Mean and standard deviation values of normally distributed nodal power demand daily proﬁle

The hourly per day operating state probability proﬁles of the EV are presented in Figure 7: p0, + p− and p correspond to the proﬁles of disconnected, charging and discharging states, respectively.

p0 p− p+

Figure 7: Hourly per day probability data of EV operating states

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Table 5: Power sources parameters and technical data [13, 17, 28, 29, 35–37]

Distributions parameters, Type j Technical parameters Costs failure and repair rates MS µ = 4000 (kW) σMS 125 (kW) P MS 4250 (kW) = MS cap = F Cov = 0.145 ($/kWh) λ = 4.00e 04 (1/h) R − λ = 1.30e 02 (1/h) − Ta = 30.00 (◦C)

NoT = 43.00 (◦C) PV Isc = 1.80 (A) α = 0.26 PV Voc = 55.50 (V) β = 0.73 Ci = 48 ($) PV F kI = 1.40 (mA/◦C) λ = 5.00e 04 (1/h) Cov = 3.76e 05 ($/kWh) R − − kV = 194.00 (mV/◦C) λ = 1.30e 02 (1/h) − VMPP = 38.00 (V)

IMPPj = 1.32 (A) W wsci = 3.80 (m/s) σ = 7.96 ws 9.50 (m/s) λF 6.00e 04 (1/h) Ci 113, 750 ($) W a = = = R − wsco = 23.80 (m/s) λ = 1.30e 02 (1/h) Cov = 3.90e 02 ($/kWh) W − − PR = 50.00 (kW) λF 2.00e 04 (1/h) Ci 17, 000 ($) EV P EV 6.30 (kW) = = R = R − λ = 9.70e 02 (1/h) Cov = 2.20e 02 ($/kWh) PST 0.28 (kW/kg) λF 3.00e−04 (1/h) Ci 135.15− ($) ST R = = = ST R − SE = 0.04 (kJ/kg) λ = 7.30e 02 (1/h) Cov = 4.62e 05 ($/kWh) − −

Coherently with constraints (16) and (17), the budget is set to BGT = 4, 500, 000 ($) and the limit of units of the different DG technologies available to be purchased is τ = [20000, 8, 250, 10000]. The maximum value of the energy price is eph = 0.12 ($/kWh) [5][5] and the highest value of total demand TLh is set to 4800 (kW). The opportunity cost for kW h not supplied Cop is considered as twice of the maximum energy price.

A total of NS = 500 random scenarios are simulated by the MCS–OPF with time step ts = 1 (h), over a horizon of analysis of 10 years (th = 87, 600 (h)), in which the investment and ﬁxed costs are pro–rated hourly.

The DE iterations are set to perform Gmax = 500 generations over five different cases of population NP 10, 20, 30, 40, 50 . The differential variation amplification factor F is 1 to maintain the integer–valued∈ { definition of the} individuals after the mutation, whereas the crossover coefficient Coc is 0.1.

HCDE runs are performed under the same conditions set for DE (Gmax , F and Coc), but for the population size NP of 50 individuals. A sensitivity analysis is performed over the HCA control parameters, namely the cophenetic correlation coefficient CCCth and linkage distances cutoff level coefficient pco, for all the nine possible pairs (CCCth, pco) with CCCth 0.6, 0.7, 0.8 and ∈ { } pco 0.25, 0.50, 0.75 . Finally, for each of the five DE and nine HCDE settings, twenty realizations are∈ carried { out. }

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4.2 Results and discussion

The results of the DE MCS–OPF for the different population sizes NP 10, 20, 30, 40, 50 are shown ∈ { } in Figure 8. The 50th percentile (%ile) or median values of the minimum global costs ECGmin, obtained from each experiment with ﬁxed values of NP, are presented as functions of the respective numbers of objective function evaluations NFE; the error bars represent the 15th and 85th %iles.

15-85th %iles 50th %ile

Figure 8: ECGmin vs NFE for NP 10, 20, 30, 40, 50 set in DE ∈ { }

(NP, CCCth, pco) 50th %ile

Figure 9: ECGmin vs NFE for each (NP, CCCth, pco) set in HCDE

As expected, for the same number of generations set in the DE MCS–OPF,the larger the population size considered the lower the values of ECGmin obtained (better ‘quality’ of the minimum). Addition- ally, we can observe marked tendencies in the reduction of both median and 15–85th %iles values of

ECGmin for increasing NFE. Performing a curve ﬁtting over these values, we get: ECGmin;50th%ile = 0.13 0.115 0.118 49.07NFE− , ECGmin;15th%ile = 49.07NFE− and ECGmin;85th%ile = 49.07NFE− , with the 2 2 2 respective coefﬁcients of determination R50th%ile = 0.994, R15th%ile = 0.998 and R85th%ile = 0.998.

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The fact that the difference between the values of the 15–85th %iles is constant indicates that the dispersion in the ECGmin(NFE) does not depend on NP and can suggest that the global searching performed by the DE is performed homogeneously in the feasible space that contains multiple local minima.

Figure 9 reports the median ECGmin values corresponding to the HCDE MCS–OPF realizations superposed to the distribution of the median ECGmin and 15–85th %iles values of the base DE experiments represented by the square markers and shaded area, respectively. The vertical and horizontal error bars account for the 15–85th %iles of the outcome ECGmin and NFE values.

Focusing on CCCth, it can be noticed that for the two extreme cases, CCCth = 0.6 and 0.8, the dispersion of the number of objective function evaluations is relatively small. On the contrary, the cases with a CCCth = 0.7 present high variability. This can be explained by the behavior of the CCC along each generation G in the evolution loop. Figure 10 shows the median, 15th and 85th %iles CCC values as a function of generation G derived from all HCDE MCS–OPF realizations. On the one hand, recalling that CCCth is used to control whether it is convenient to perform HCA, the small NFE dispersion in the case with CCCth = 0.6 is because clustering is practically been G applied in all generations (CCCth CCC ), thus disabling any effect generated by passing from populations with original size NP to≤ reduced populations with NPG NP and vice versa. On the ≤ other hand, the effect is also being avoided in the case CCCth = 0.8 by not applying clustering. Indeed, in Figure 10 it can be observed that after the generation 50 it is unlikely that by performing HCA the proposed hierarchical grouping structures represent well enough the population.

50th %ile 15-85th %iles

Figure 10: CCC behavior per generation G

Differently, the cases for which CCCth = 0.7 present high dispersion in the NFE since the median values of CCC G move in the neighborhood of the threshold throughout the major part of the evolution loop in the HCDE. Moreover, in general terms, the values of CCC G 15–85th %iles

20 158 maintain certain symmetry with respect to the median, i.e., performing or not HCA are equally likely events, producing high ﬂuctuations in the number of individuals considered as population and, therefore, affecting in the same way the NFE.

The above mentioned insights are noticeable also in Figure 11, which shows the empirical probability density functions (pd f s) of the population size NPG per generation for each (NP,

CCCth, pco) set in HCDE. Indeed, the average probabilities of performing HCA throughout the evolution cycle for the different values of CCCth = 0.6, 0.7 and 0.8 are 0.980, 0.540 and 0.078, respectively.

(50, 0.6, 0.25) (50, 0.7, 0.25) (50, 0.8, 0.25) (50, 0.6, 0.50) (50, 0.7, 0.50) (50, 0.8, 0.50) (50, 0.6, 0.75) (50, 0.7, 0.75) (50, 0.8, 0.75)

G Figure 11: Empirical NP pd f for each (NP, CCCth, pco) set in HCDE

Regarding the cutoff level coefficient of the linkage distances pco, in Figure 11 it is possible to identify the three peaks of reduction in the population size, confirming the role of this control parameter in defining the level at which the hierarchical structures proposed are ‘cut off’ when the

From the results obtained for all the different DE and HCDE settings, we look for six representative cases for the analysis (Figure 9). From the DE runs, we select the settings with extreme and middle population size NP 10, 30, 50 , whereas from HCDE we choose the cases (NP, CCCth, pco) set as (50, 0.6, 0.25), (50,∈ 0.6, { 0.50),} (50, 0.7, 0.50) and (50, 0.7, 0.75). The former (50, 0.6, 0.25) and (50, 0.6, 0.50) cases present signiﬁcant reductions in the number of NFE, with small dispersion and loss of quality of the minimum ECG obtained, compared to the results obtained by diminishing directly the ﬁxed NP in DE from 50 to 10. Similarly, the cases (50, 0.7, 0.50) and (50, 0.7, 0.75) may lead to considerable reductions in NFE, with acceptable losses of ECGmin, but subject to a high degree of variability that compromises the performance.

21 159

As for computational times, running on an Intel® Core™i7–3740QM (PC) 2.70 GHz without performing parallel computing, the average time to evaluate the objective function is 4.592 (s) for the NS = 500 scenarios in the MCS–OPF; for a ﬁxed population of NP = 50 and its corresponding NFE = 20, 050, the total time for a single run is on average 25.574 (h). Taking into account this, under commonly used hardware conﬁgurations, the reductions in computational time that can be achieved by using HCDE with (50, 0.6, 0.25) and (50, 0.6, 0.50) settings are 19% and 49% for the median, 23% and 51% for the 15th %ile, and 16% and 43% for the 85th %ile, respectively.

The integration of HCA into the DE algorithm introduces a signiﬁcant time complexity, conditioning the reductions of computational efforts that can be obtained by applying the proposed HCDE MCS–OPF framework. Indeed, if performing HCA along all generations of DE and running the

MCS–OPF on an eventually reduced population (depending on CCCth and pco) is computationally heavier than running the MCS–OPF over the complete population, the effects of the framework can be negligible or even negative. It is possible to formulate the condition to obtain reductions in the computational efforts by the proposed HCDE MCS–OPF framework, from the asymptotic time complexities of the main algorithms that compose it. Table 6 reports the independent asymptotic time complexities as functions of the generic size m of the input to each algorithm and of the parameters that deﬁne the dimensionality of the HCDE MCS–OPF framework [26, 38].

Table 6: Asymptotic time complexity of the algorithms

Algorithm PDIST HC MCS OPF Time complexity T 2 a 2 b O(dm ) O(m log(m)) O(m) O(size(A)) 2 2 2 O(nps NP ) O(NP log(NP)) O(NS nps) O(NS nps ) × × × a Pairwise distance PDIST between all m vectors of size d. b The matrix A comes from the canonical form Ax b of the linear programming of the DC OPF problem approximation. ≤ where nps represents the size of the DG–integrated network, i.e., the number of nodes n times the number of all the technologies of power generation available ps, NP is the size of the complete population and NS is the number of scenarios in the MCS–OPF.

Comparing the asymptotic time complexities of the algorithms involved in the realization of the proposed framework with and without integrating HCA, the following inequalities must be fulﬁlled in order to obtain a reduction in the computational time by HCDE:

PDIST HC G MCS OPF MCS OPF T (nps, NP) + T (NP) + E[NP ] T − (NS, nps) < NP T − (NS, nps) × × 2 2 ⇓G 2 2 nps NP + NP log(NP) + E[NP ] NS nps < NP NS nps (26) × × × × × NP NP log NP ⇓ E NP G κ ( ) ε < 1 n, ps, NP, NS , ε [ ] 0, 1 = + 2 + Z∗ = ( ] NP nps NP nps ∀ ∈ NP ∈ × × 22 160 where ε is the expected ratio of the population NPG evaluated along all generations G of DE to the total population NP and κ is the ratio of the asymptotic time complexities of HCDE to DE.

From Equation (26), we can observe that the contribution of the terms related with the complexity of MCS–OPF, dependent on NS and nps, is considerably large for the fulfillment of the inequality conditions. In fact, when using DE, it is commonly accepted to set a size of the population NP not greater than ten times the size of the decision variables, in this case, 10nps [24], making the first two terms of κ strongly dependent on the number of scenarios NS. Moreover, given the complexity of the general problem, higher values of NS lead to a better approximation of the objective function via MCS–OPF, i.e., the more likely is to fulfill the condition and the greater can be the reduction of computation time. However, the value of ε depends on the probability of performing clustering in each generation and at what level, controlled by CCCth and pco respectively. In some cases, ε can be close to 1 (as we inferred from Figure 11) implying negligible benefits. Table 7 shows the values of the ratio κ for each (NP, CCCth, pco) set in HCDE considering the dimensionality of the present case study defined by the values of the parameters nps = 55, NS = 500, NP = 50. The value of 1 κ can be interpreted as the expected asymptotic relative time reduction achieved by performing HCDE.−

Table 7: Ratio κ for each (NP, CCCth, pco)

G NP NP log(NP) E[NP ] (NP, CCCth, pco) ε = κ 1 κ NP nps NP nps2 NP − (50, 0.6, 0.25) × × 0.817 0.819 0.181 (50, 0.7, 0.25) 0.921 0.923 0.077 (50, 0.8, 0.25) 0.987 0.989 0.011 (50, 0.6, 0.50) 0.510 0.512 0.488 (50, 0.7, 0.50)1.818e 03 3.418e 05 0.738 0.740 0.260 (50, 0.8, 0.50)− − 0.978 0.979 0.021 (50, 0.6, 0.75) 0.259 0.261 0.739 (50, 0.7, 0.75) 0.487 0.488 0.512 (50, 0.8, 0.75) 0.909 0.911 0.089

Figure 12 shows the convergence curves for the DE and HCDE cases selected, for the twenty runs performed for each (NP, CCCth, pco) setting: no signiﬁcant differences can be found among the convergence curves except for the expected behavior of converging to lower values of ECGmin for settings which imply a larger population size.

Figure 13 shows the average total DG power allocated in the distribution network and the corresponding investment costs of the DE and HCDE MCS–OPF cases selected, choosing the corre- DG sponding optimal DG–integrated plans as the decision matrices Ξ for which their ECGmin values are the closest to the median ECGmin value obtained for the twenty runs of each (NP, CCCth, pco) setting. It can be pointed out that in all the cases, the contribution of EV is practically negligible if compared with the other technologies. This is due to a combination of two facts: the probability

23 161 that the EV is in a discharging state is much lower than that of being in the other two possible operating states, charging and disconnected (see Figure 7) and when EV is charging, the effects are opposite to those desired, i.e., it is acting as loads.

(10, ˗, ˗) 15-85th %iles (50, 0.6, 0.50) 15-85th %iles (30, ˗, ˗) 15-85th %iles (50, 0.6, 0.25) 15-85th %iles (50, ˗, ˗) 15-85th %iles (50, 0.6, 0.50) 50th %ile (10, ˗, ˗) 50th %ile (50, 0.6, 0.25) 50th %ile (30, ˗, ˗) 50th %ile (50, ˗, ˗) 50th %ile

(50, 0.7, 0.50) 15-85th %iles (50, 0.7, 0.75) 15-85th %iles (50, 0.7, 0.50) 50th %ile (50, 0.7, 0.75) 50th %ile

Figure 12: Convergence curves for representative (NP, CCCth, pco) settings

In all generality, both the investment cost Ci and the average power installed by DG is comparable in all the cases, except for the setting (50, 0.7, 0.75) for which the level of clustering determined by pco = 0.75, that translates into higher reductions of the population size, may lead to less similar local minima than the other settings.

The average total renewable DG power allocated per node is summarized in Figure 14. Even though all the ECG optimal decision matrices ΞDG show differences, the tendency is to install localized sources of renewable DG power between two identiﬁable portions of the distribution network, up and downstream the feeder (2, 6) (Figure 5), giving preference to the second portion which presents higher and non–stream homogeneous nodal load proﬁles.

24 162 h osblt ffiue fntokcmoet r nlddi ot al iuain which simulation, Carlo Monte a as in well included as are loads, components and network supply of uncertain power failures main inherent of the The possibility in the system. variability the sources, of energy cost renewable global of expected behavior the with considered minimizing is of optimization of objective The planning the network. the respect distribution for a framework into optimization generation and renewable simulation of a integration presented have we paper, previous a In Conclusions 5 ape elztoso h neti prtoa cnro o h pia oe ﬂow. power optimal the for scenarios operational uncertain the of realizations samples iue13: Figure vrg oa Gpwralctdadivsmn otfrrpeettv ( representative for cost investment and allocated power DG total Average iue14: Figure oa vrg oa Gpwrfrrpeettv ( representative for power DG total average Nodal 1 2 node i 3 4 5 6 7 8 9 10 25 11 P N , CCC Ci ($) th , P N p co , settings ) CCC th , p co settings ) 163

The framework is quite general and complete in the characteristics of the realistic system scenarios considered. However, this is at the expenses of the computational time required for the overall optimization.

In this respect, in the present paper we have addressed the problem of computational efﬁciency in the resolution of the renewable DG planning optimization problem. We have done so by an original introduction of a controlled clustering strategy, with, the main original contributions being:

• The integration of differential evolution and hierarchical clustering analysis for grouping similar individuals from a given population and selecting representatives to be evaluated for each group, thus reducing the number of objective function evaluations during the optimization.

• The introduction of two control parameters, namely the cophenetic correlation coefﬁcient and a cutoff level coefﬁcient of the linkage distances, for allowing controlled adaptation during the search process and decision on whether or not to perform clustering and at which level of the hierarchical structure built.

A case study has been analyzed derived from the IEEE 13 nodes test feeder. The results obtained show the capability of the framework to identify optimal plans of renewable DG integration. The sensitivity analysis over the control parameters of the hierarchical clustering shows that the efﬁciency is improved with cophenetic correlation thresholds that allow the clustering in almost all generations along the differential evolution, setting the level of clustering to no more than the ﬁfty percent of the feasible linkage distances range in the hierarchical structure proposed. Indeed, this is shown to lead to acceptable reductions in the number of objective function evaluations, with small dispersion and loss of quality in the minimum global cost obtained.

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29 Paper (iii)

R. Mena, M. Hennebel, Y.–F. Li and E. Zio, “A multi-objective optimization framework for risk–controlled integration of renewable generation into electric power systems,” Energy, (under review) 168

A Multi-objective Optimization Framework for Risk-controlled Integration of Renewable Generation into Electric Power Systems

Rodrigo Menaa, Martin Hennebelb, Yan-Fu Lia, Enrico Zioa,c,1

aChair on Systems Science and the Energetic Challenge, Fondation Electricité de France at CentraleSupeléc, Chatenay-Malabryˆ Cedex, France bCentraleSupeléc, Department of Power & Energy Systems, Gif-Sur-Yvette, France cPolitecnico di Milano, Energy Department, Milan, Italy

Abstract

We introduce a multi–objective optimization (MOO) framework for the integration of renewable distributed generation (DG) into electric power networks. The framework searches for the optimal size and location of different DG technologies, taking into account uncertain ties related to primary renewable resources availability, components failures, power demands and bulk–power supply. A non–sequential Monte Carlo simulation and optimal power ﬂow (MCS–OPF) computational model is developed to emulate the network operation by generating random scenarios from the diverse sources of uncertainty, and assess the system performance in terms of global cost (CG). To measure uncertainty in the system performance, we introduce the conditional value–at–risk deviation (DCVaR) which, due to its axiomatic relation to the CVaR, allows the conjoint control of risk. A MOO strategy can, then, be adopted for the concurrent minimization of the expected global cost (ECG) and the associated deviation DCVaR(CG). In our work this is operatively implemented by a heuristic search engine based on differential evolution (MOO–DE). An example of application of the proposed framework is given with regards to the IEEE 30 bus test system, where in DCVaR is shown capable of enabling and controlling tradeoffs between optimal expected economic performance, uncertainty and risk.

Keywords: Renewable distributed generation, uncertainty, risk, optimization, differential evolution, conditional value–at–risk, conditional value–at–risk deviation.

1 Corresponding author. Tel: +33 1 4113 1606; fax: +33 1 4113 1272. E-mail addresses: [email protected], [email protected] (E. Zio).

1 169

1 Introduction

The integration of renewable distributed generation (DG) into electric power systems is playing a relevant role in the strategies to decentralize and diversify the overall power generation and, in the efforts to contribute to the pursuit of environmental sustainability. In principle, the allocation of renewable DG units directly on ‘the customer site of the meter’ and/or on strategically–defined points of the sub–transmission grids can improve reliability of power supply, alleviating the restrictive dependence on the bulk power generated by large–scale conventional power plants [1, 2]. Furthermore, improvements of power losses and voltage stability profiles can be achieved, as well as reductions of investment risks and transmission costs [2–4]. DG planning is subject to complex economic, regulatory, technical and operational constraints, which must be attentively considered in order not to generate complications in the operative power system that may end up counteracting DG’s potential benefits [5–7]. One of the main difficulties associated to the selection of technologies, dimensioning of power capacities and definition of the location of the different renewable generation units, is the modeling of the intrinsic uncertain behavior of the primary renewable energy sources (e.g. solar irradiance, wind speed and water inflow, in the case of solar photovoltaic, wind turbines and hydro–power technologies, respectively) and of the stochastic occurrence of unexpected events on the power generation units, such as failures and stoppages. Indeed, these sources of uncertainty are adjoined to those already present in the operative power system: failures and stoppages of transmission and distribution (T&D) lines and conventional generators, variability in the power demands and energy prices, fluctuations in the bulk power supply, etc. Consequently, for any proposed plan of DG–integrated network, the stochastic operational conditions need to be emulated coherently to the various sources of uncertainty to assess its expected performance by solving power flow equations [1, 8–10]. The selection of DG–integrated network plans can be framed as an optimization problem, whose complexity is driven by the size of the network and number of available DG technologies, that can lead to combinatorial explosion [9,11,12] and, by the aforementioned stochasticity and uncertainty. Optimality of the plan is customarily sought with regards to economic targets, such as minimization of costs of carbon dioxide emissions, fuel and transportation, energy losses, operation and maintenance, investment, etc.. The solutions identified must, of course, comply with technical constraints like generation capacities, T&D lines rating and voltage drops [1, 5]. Evolutionary algorithms (EAs) have emerged as the most effective search engines for combinatorial optimization problems and they can deal with non–differential objective functions, discontinuous feasible spaces and non–convex conditions [9,11]. Some of the best known techniques are: particle swarm optimization (PSO) [9, 13–16], differential evolution (DE) [17–19] and genetic algorithms (GA) [2, 12, 20–22]. The uncertainty in the physical parameters and variables of a DG–integrated network propagates

2 170 onto the uncertainty in its operational response and the risk of incurring in non–desirable outcomes or non–satisfactory performance. This demands a framework of evaluation and control, for allowing properly informed and confident decision making. For evaluation purposes, various deviation and risk measures have been introduced into the DG planning optimization frameworks, framing the problem as a portfolio optimization in which the different types of DG technologies are treated analogously to financial assets [9, 23–30]. In portfolio optimization theory, variance is the most commonly used deviation measure for estimating uncertainty, although other indicators, like the mean absolute deviation (MAD), have also been employed [31]. Considering a probabilistic performance function associated to a certain portfolio, which usually is defined as a return (or loss) function, its variance is considered as an indicator of the likelihood of achieving the mean value of performance of the portfolio. This mean–variance approach can be applied to various portfolio optimization strategies, like the (single– objective) minimization of mean loss (maximization of mean return) with constrained variance, the (single–objective) minimization of variance with constrained mean loss or the simultaneous (multi–objective) minimization of mean loss and variance which enables the investors to select (an) optimal portfolio(s) by trading–off or conditioning mean loss and uncertainty (variance) on the Pareto front of dominant solutions. Mean–variance approaches have been applied to DG planning [9, 23, 24, 27, 30, 32] even though, one important warning must be considered from a risk perspective: the variance measure includes symmetrically the values of performance that fall short of or exceed the mean value; in the search for optimal DG technologies portfolios, lower levels of uncertainty (variance) in the performance function can be obtained with portfolios that lead to rarer occurrences of both beneficial and/or non–desired (risky) scenarios. Then, for controlling the risk side, it is necessary to introduce additional indicators that provide information on the extent of asymmetry of the performance function, weighting accordingly the risky part of it. For instance, skewness and kurtosis indicators [33] have been traditionally integrated into mean– variance approaches, for both financial and engineering applications, to estimate the asymmetry and peakedness of a probabilistic performance function, respectively, and allow controlling conjointly mean performance, uncertainty of it and risk.

Similarly to mean–variance approaches, and likewise derived from portfolio optimization theory, direct risk–based frameworks have been formulated and applied to tackle DG planning problems under uncertain conditions. As for mean–variance approaches, similar optimization strategies can be implemented to target expected performance and risk separately, as objectives or constraints, or to search for an efficient Pareto front of them. For this, the most widely used risk measures are the value–at–risk (VaR) and conditional value–at–risk (CVaR) [22,26,28,29,34–37], both with focus on the non–desirable performance outcomes given by a portfolio relative to a specific confidence level or percentile: VaR is defined as the threshold value of performance at the confidence level, whereas CVaR is the expected value of the performances that fall beyond the VaR [38]. For optimization under uncertainty, CVaR has gained more interest than VaR because of its preferable properties: among

3 171 others, CVaR maintains consistency with VaR and is a coherent risk measure; it can be expressed by a minimization formula and aims towards conservatism, which prevails in risk management [38]. Even though integrated portfolio optimization approaches present robustness advantages by conjointly controlling the level of uncertainty and risk associated to the mean value of performance of different portfolios, they can considerably increase the complexity of the concurrent optimization problem that, in our case of interest, is already complicated given its combinatorial nature due to the size of a DG–integrated network, and the presence of uncertainty in the operational conditions. Considering the mean value and the necessary deviation and risk measures as representative objectives of a portfolio increases signiﬁcantly the number of objective functions to be simultaneously optimized, further constraining the feasible space and, eventually, hindering in understanding the information delivered to the decision–makers.

In this work, we take the challenge of developing an optimization framework for the integration of DG into a electric power network accounting for uncertain operational inputs, and assessing and controlling uncertainty and risk. For this, we exploit some relevant concepts and tools from portfolio optimization, but avoiding increasing the complexity of the problem by adopting suitable performance indicators and, eventually, providing a spectrum of comprehensible information for well–supported decision making. The main original contribution of this work lies in the introduction of the CVaR deviation (DCVaR) [31] as a measure of uncertainty, used along with the expected value as performance indicators of the global cost function (CG) associated to each portfolio of DG technologies. A multi–objective optimization (MOO) strategy is, then, developed, aiming at the simultaneous minimization of both objective functions: the expected global cost (ECG) and its corresponding DCVaR. The numerical implementation of the approach is based on a Monte Carlo–optimal power ﬂow (MCS–OPF) simulation model nested in a MOO search engine based on differential evolution (DE) [39]. The MCS–OPF model emulates the operation of each DG–integrated network proposed by MOO–DE, generating random scenarios from the diverse uncertain operational inputs and assessing the global cost performance CG of each, and, then, evaluating the two objective functions. The proposed framework searches for optimal technologies, size and location of DG units, and enables the direct trade–off between optimal expected performance and the associated uncertainty to achieve it. Furthermore, thanks to the axiomatic relation between CVaR and DCVaR, it integrates in the optimal Pareto front the level of risk, given by the values of CVaR.

We apply the framework on a case study based on the IEEE 30 nodes test feeder sub–transmission and distribution network [40], considering solar photovoltaic and wind turbines DG technologies, discussing the effectiveness and implications of of its practical application.

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2 DG–Integrated power network simulation model

In this section, we present the Monte Carlo and optimal power ﬂow (MCS–OPF) simulation model. We introduce the representation of the DG–integrated network, the modeling of the different uncertainty sources considered, the process of generating the random operational scenarios and the formulation of the OPF problem.

2.1 Network conﬁguration

For modeling a DG–integrated network, one needs the definition of the type (or classes) of components involved and their topological location. In this work, we consider three main classes of components: power generators G, transmission and distribution lines TD and power demands (loads). The class of power generators considers all the different types of conventional bulk power suppliers MG and renewable technologies RG. The topology of the network is described as a graph, i.e., a set of nodes and the various connections that link them. The nodes represent spatial points at which generation components (MG and RG) and loads are located or can be allocated, whereas the connections between nodes are the transmission and distribution lines. We indicate a node by the index i and the set of all nodes by N = i : i 1, 2,..., n , where n is the total number of nodes in the network. Consequently, we define{ the∈ set { of transmission}} and distribution lines as Y = (i, i') : nodes i and i' are connected, i, i' N . { ∀ ∈ } Assuming stationary operational conditions, the network performance is considered dependent on the fixed location and magnitude of the power available in each generation unit and loads and, the technical limits of the T&D lines. To indicate the location and capacity size of the different types of generation units present in the network, we define the following matrix Q, i N as follows: ∀ ∈

m + r types of power generators G m types of bulk power suppliers MG r types of renewable technologies RG

G1 Gj Gm G1+m Gj+m Gr+m ··· ··· ··· ··· q1,1 q1,j q1,m q1,1+m q1,j+m q1,r+m . ···. . ···. . . ···. . ···...... (1) M R   Q = [Q Q ] = qi,1 qi,j qi,m qi,1+m qi,j+m qi,r+m |  . ···.. . ···.. . . ···.. . ···.. .   ......  q q q q q q   n,1 n,j n,m n,1+m n,j+m n,r+m ··· ··· ··· ··· where [Q]i,j = qi,j is a non–negative integer that speciﬁes the number of units of the power generator type j allocated at node i:

q' Z∗ if q' units of Gj are allocated at node i, Gj G = G1,..., Gm+r qi,j = ∈ ∀ ∈ { } (2)  0 otherwise 

 5 173

Then, the static configuration of the physical components G and TD of a DG–integrated network is represented by the pair ([Q], Y ). Any physical component is assumed to be affected by the stochastic occurrence of failures,{ conditioning} dynamically the functionality of power generators and the paths through which the power flows. Moreover, the magnitude of power available in each generator is subject to the inherent uncertain behavior of the corresponding primary energy source and, considering the DG–integrated network as a ’price taker’ entity, the economic conditions depend on the variability of the power demands [41, 42]. Hence, to evaluate the operating performance of a given DG–integrated network, affected by significant uncertain conditions, it is essential to model the different sources of variability and emulate the response of the network for a large representative combination of possible scenarios.

2.2 Uncertainty modeling

Analytical methods and Monte Carlo simulation (MCS) are among the most common techniques for evaluating the performance of DG–integrated networks [2]. In theory, analytical methods are preferable because of the possibility of achieving closed form solutions; however, their application often requires simplifications in the modeling which may lead to unrealistic results. For instance, analytical solutions for optimal DG planning consider non–uncertain or non–intermittent power generation and/or load profiles, and networks of low dimensionality [43]. On the contrary, MCS allows a more realistic modeling, because the performance of the network is not analytically solved but simulated, and the overall performance indicators are statistically estimated from virtual operational scenarios realizations [44]. MCS has been found quite adequate for the analysis of distribution networks with a significant number of sources of randomness or variability, e.g., power generation, loads, component failures or degradation processes, etc. [2, 8, 9, 16, 27, 28], but at the expense of incrementing the use of computational resources.

In the present framework, we adopt a non–sequential MCS, based on latin hypercube sampling (LHS) [45], to emulate the operation of the DG–integrated network, considering the operation variables as independent on previous uncertain conditions, so as to seize the advantages of MCS without overly increasing the computational efforts.

The considered uncertain conditions that determine the operation of the DG–integrated network are accounted for using different stochastic models, as presented below.

2.2.1 Power demands

The aggregated profile of power demand in an electric power network, as well as the single nodal profiles, can be represented as daily load curves inferred from historical data [42, 46], and can be considered uncertain following normal distributions [9,16]. We model the nodal power demand profiles by integrating both ideas, i.e., for a specific hour of the day t D = 1,..., 24 the ∈ { } 6 174

corresponding power demand at node i, denoted Li,t (MW), is normally distributed with mean µi,t

(MW), standard deviation σi,t (MW) and truncated at 0:

φ(ξ(Li,t , µi,t , σi,t )) L , µ , σ 0 σ Z µ , σ i,t i,t i,t fi,t (Li,t µi,t , σi,t ) = i,t ( i,t i,t ) ∀ ≥ (3) |   0 otherwise L µ µ  i,t i,t i,t ξ(Li,t , µi,t , σi,t ) = − ; Z(µi,t , σi,t ) = 1 Φ σi,t − σi,t where, φ and Φ are the standard Normal probability density function and its cumulative distribution function, respectively.

As aforementioned, the network is assumed as a ’price taker’ entity, for which the value of the energy price is correlated with the aggregated power demand in the network. As an intermediate approximation of existing studies (e.g. [41, 42, 47]), the proportional correlation used in this study can be expressed as:

L L EP L EP , L EP i,t 0.38 i,t 1.38 (4) t ( i,t max maxi ) = max L L + | i N maxi − i N maxi X∈ X∈ where, EPt ($/MWh) is the energy price at the hour of the day t and EPmax ($/MWh) is the maximum value of energy price correspondent to the aggregated value of maximum nodal power loads L (MW). maxi

2.2.2 Bulk power generation

Bulk power generation stands for the power supply coming from conventional power plants (MG) already existing in the network. These sources of power supply are rather stable and are connected to the network at sub–transmission or distribution transformers. Their stochastic behavior is represented following normal distributions [11, 48], with small standard deviation and truncated by the maximum capacity of generation.

φ(ξ(Pj, µj, σj)) P 0, P , µ , σ 0 j [ max j ] j j f P µ , σ , P σj Z(µj, σj) (5) j( j j j max j ) = ∀ ∈ ≥ |   0 otherwise P µ P µ µ j j max j j j ξ(Pj, µj, σj) = − ; Z(µj, σj) = Φ( − ) Φ( ) σj σj − σj where j j : G MG , P (MW) and P (MW) are the available bulk power and maximum j j max j ∀ ∈ { ∈ } capacity of the MG generator type j, respectively, and µj (MW) and σj (MW) the corresponding Normal distribution mean and standard deviation.

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2.2.3 Solar photovoltaic generation

Solar photovoltaic technologies (PV) transforms solar irradiance into electric power through panels of solar cells. Commonly for long term periods of analysis, solar irradiance uncertain behavior has been modeled using probabilistic distributions, obtained from weather historical data of a particular geographical area. In particular, the Beta distribution function has been found suitable to model hourly solar irradiance [46,49] and, therefore, is adopted in this paper. Moreover, the intermittency in the solar irradiation is taken into account deﬁning a daylight interval between 07.00 and 21.00 hours, i.e., if the value t of the hour of the day is in the subset of DL = 7,..., 21 of D, the solar { } irradiation H is a positive value, otherwise t is in the night interval DN = 22,..., 24, 1,..., 6 and the value of solar irradiation is assumed equal to 0. Then, we adjust the Beta{ distribution function} considering the probability that t is in the daylight interval pL = P(t t DL), as follows: | ∈ α 1 ( i ) (βi 1) Γ (αi + βi) Hi − (1 Hi) − − Hi H , 1 , αi, βi > 0 Γ α Γ β 1 p B α , β [ i∗ ] fi(Hi αi, βi, pL, Hi∗) = ( i) ( i) ( L) ( i i) ∀ ∈ (6) |  −  0 otherwise

 where j j : Gj PV , Hi is the solar irradiance at node i, αi and βi are the shape parameters ∀ ∈ { ∈ } of the corresponding Beta probability density function at node i and Hi∗ is the pL percentile of the non–adjusted Beta distribution fi,j(Hi αi, βi). | Given the technical characteristics of PV cells and the model of solar irradiance, the probabilistic power output of a single cell is obtained from the following equation:

P' H if 0 P' H P i,j( i) i,j( i) max j Pi,j(Hi) = ≤ ≤ (7)  Pmax if Pmax < P'i,j(Hi)  j j P' H n FF V H I H 10 6 i,j( i) = c j ( i) ( i) − × T H T H T 20 /0.8 C ( i) = Ai + i( N oj + ) I T H I k T 25 ( C ) = i( SCj + I j ( C )) − V T V k T ( C ) = OCj + Vj C FF V I / V I j = ( MPPj MPPj ) ( OCj SCj ) where referring to PV type j, P (MW) is the power output at node i, P (MW) is the maximum i,j max j power generation capacity, n is the number of photovoltaic cells, FF is the fill factor, T ( C) is c j Ai ◦ the ambient temperature at node i, T ( C) is the nominal cell operation temperature, I (A) is N oj ◦ SCj the short circuit current, k (mA C) is the current temperature coefficient, V (V) is the open I j /◦ OCj circuit voltage, k (mV C) is the voltage temperature coefficient, and V (V) and I (A) are Vj /◦ MPPj MPPj the voltage and current at maximum power, respectively.

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2.2.4 Wind turbines generation

β U (βi 1) U βi i i − exp i U 0, α , β > 0 α α α i i i fi(Ui αi, βi) =  i i − i ∀ ≥ (8) |  0 otherwise  where j j : Gj W , Ui (m/s) is the wind speed at node i and αi and βi are the scale and shape parameters∀ ∈ { of∈ the} Weibull distribution function at node i, respectively.

Similarly to PV type of technologies, the uncertainty associated to the wind speed and the technical features of a speciﬁc type of wind turbine characterize its power output function that can be determined as follows:

U U i CI j − P if U U < U U U R j CI j i Aj Aj CI j ≤  − Pi,j(Ui) = P if U U U (9)  R j Aj i COj  ≤ ≤ 0 otherwise   where referring to W type j, P (MW) is the rated power and U , U and U (m s) are the R j CI j Aj COj / cut–in, average and cut–out wind speeds, respectively.

2.2.5 Components availability state

Consistently with the non–sequential nature of the MCS simulation proposed in the present framework, the availability states of the physical components in the network, generators and T&D lines, are straightforwardly modeled by two–state stationary Markov chains [2, 50], deﬁning two possible operating states: η = 0 if the corresponding component is non functional (failure) and η = 1 if the component is available to operate, i.e., generate, transmit or distribute power accordingly to the class of component. Then, the discrete stationary distribution of operating states can be expressed as follows: λ / λ λ η 0 Fk ( Fk + Rk ) k = fk(ηk λF , λR ) = (10) k k  | λR /(λF + λR ) ηk = 1  k k k where k k : G G k/k i, i' Y , η is the operating state of component k and λ k = ( ) k Fk and λ ∀are∈ {{the corresponding∈ } ∪ { failure and∈ repair}} rates, respectively. Rk

9 177

2.3 Monte Carlo and optimal power ﬂow simulation

For a given DG–integrated network, represented by the pair ([Q], Y ) that contains the locations and number of units of the different power generators and the T&D{ lines,} each uncertain variable is randomly sampled several times by LHS [45] and the inverse transform method [44], for the realization of NS operational scenarios of duration t∆. For practicality, we deﬁne NS as multiple of 24, so each hour of the day has the same number of realizations NS/24. We denote by ω the set of sampled variables, which constitutes an operational scenario and by Ω the sets of all the NS realizations of ω.

ω t , L , EP , P /G MG , H , U , η , η (11) s = s i,ts ts i,j,s j i,s i,s i,j,s (i,i'),s { { ∈ } } Ω = ws : s 1, 2, . . . , NS (12) { ∈ { }}

For each scenario ωs, the configuration ([Q], Y ) and the sampled variables set the stage for evaluating the response of the network in terms of available{ } power usage, power demand satisfaction and the involved economics. For this, DC optimal power flow analysis (OPF) is run, neglecting power losses, assuming constant the voltage throughout the network and accounting solely for active power flows [51, 52]. The OPF receives ([Q], Y ) and ωs as inputs and aims the minimization of the aggregated operating cost of generation,{ transmission} and distribution and load shedding, including revenues per MW h sold. The present formulation of the power flow problem is:

MCS–OPF([Q], Y , ω): { } min COω COv EP P s = ( j ts ) Ui,j,s + PUs ,∆δs,LSs i N Gj G − X∈ X∈ (13) C EP LS S COv B δ δ ( LS + ts ) i,s + re f (i,i') (i,i')( i,s i',s) i N (i,i') Y | − | X∈ X∈ s.t.

L P S η B δ δ LS 0 (14) i,ts Ui,j,s re f (i,i'),s (i,i')( i,s i',s) i,s = − Gj G − i' N − − X∈ X∈ 0 P η Q P (15) Ui,j,s i,j,s[ ]i,j i,j,s ≤ ≤ S B δ δ P (16) re f (i,i')( i,s i',s) max i,i' | − | ≤ ( ) ω where t D, s 1, 2,..., NS , COs ($/h) is the aggregated operating cost of generation, trans- ∀ ∈ ∈ { } mission and distribution, and load shedding, COvj ($/MWh) is the variable operating cost of the power generator j, EP ($ MWh) is the energy price at hour t, P (MW) is the used power from ts / Ui,j,s the generator type j located at node i, Sre f (MV A) is the reference apparent power in the network,

COv(i,i') ($/MWh) and B(i,i') (p.u.) are the variable operating cost and susceptance of the T&D line (i, i'), respectively, δi,s is the voltage angle at node i, CLS ($/MWh) is the load shedding cost and

10 178

P (MW) is the power rating of the T&D line i, i' . The load shedding LS (MW) at node i is max(i,i') ( ) i,s deﬁned as the amount of load disconnected to alleviate congestion in the feeders and/ or balance the demand of power with the available power supply.

Equation (14) corresponds to the power balance at node i, whereas equation (15) represents the bounds of the power generation units and equation (16) considers the technical limits of the feeders.

As above–mentioned, OPF is solved for each operational scenario ωs, giving in output the ω Ω ω ω ω respective values of minimum COs . The set CO = CO1 , CO2 ,..., CONS is, then, considered as a sample of realizations of the probability function of{ CO. }

2.3.1 DG–integrated network performance evaluation

M R The proposed renewable DG–integrated network solutions Q = [Q Q ] are evaluated with respect to performance indicators of the global cost CG function. The quantity| CG is composed of two terms: the outcome operating cost of the MCS–OPF described in the previous section, COΩ, and the

ﬁxed investment and operating cost, CI j + CO f j, associated to the renewable part of the proposed R DG plan Q , i.e., j j : Gj RG . The quantity CI j + CO f j ($) is prorated hourly over the lifetime of the project∀ ∈t {h. Thus,∈ the} global cost function for the set of operational scenarios Ω is given by: 1 CGΩ COΩ CI CO f QR (17) = + t H ( j + j)[ ]i,j i N Gj RG X∈ X∈ Analogously to COΩ deﬁnition, CGΩ represents a sample of realizations of the probability function of CG and performance indicators of interest can be obtained, relative to expected performance, uncertainty and risk.

2.4 Uncertainty and Risk Assessment

The proposed MOO framework introduces the CVaR deviation (DCVaR) [31] to measure the uncertainty in the performance function (CG) of interest. The quantity DCVaR is a functional of the CVaR [38], which is a coherent risk measure broadly used in ﬁnancial portfolio optimization and has been extended to engineering applications, including electric power systems analysis and, in particular, DG planning [22,26,28,29,34–36]. The axiomatic relation between DCVaR and CVaR allows optimizing simultaneously uncertainty and risk, by targeting just one of these two indicators.

The deﬁnitions and properties of CVaR and DCVaR for continuous and discrete general return (loss) functions are given in detail in [31,38]. Here, we limit ourselves to presenting only a graphical, but comprehensive view to understand the CVaR and DCVaR deﬁnitions, as shown in Figure 1.

11 ([ h orsodn assessed corresponding The Thus, n tcnb rvdta n–ooerlto xsswt t orsodn eito measure deviation corresponding its with exists relation one–to–one a DCVaR that proved be can it and account into taking defined deviation is standard It the of value. properties expected important its some exceeding loss the to associated uncertainty equations: quantities following The the by loss. expressed of scenarios risky or non–desirable extreme to equal or than greater is value threshold that exceed not does loss the that probability or CVaR Q α ] ntepeetfaeok ecncnie pcfi ofiuaino h Gitgae network DG–integrated the of configuration specific a consider can we framework, present the In measure risk expectation–bounded strictly a is CVaR measure, risk coherent a being Furthermore, to regards With loss the of function probability the of approximation discrete a For , pretl,tevalue–at–risk the -percentile, { α CVaR Y ( [ } x 31 ) ) sagnrto otoi,i hc h eeal part renewable the which in portfolio, generation a as ] steepce au fls ie httels sgetrta reult the to equal or than greater is loss the that given loss of value expected the is : α ( x iue1: Figure ) rvdsaqatttv niaino h xeto h rbblt focrec of occurrence of probability the of extent the of indication quantitative a provides DCVaR rpi ersnainof representation Graphic CG α frequency ( x Ω ) hsi o ymti eito esr,a tacut o the for accounts it as measure, deviation symmetric non a is this , bandfo h uptMCS–OPF( output the from obtained , CVaR DCVaR CVaR VaR VaR E α α α ( ( α ( x ( α ) x x x ( ( DCVaR = ) x = ) = ) x ) = ) VaR ersnstesals au fls o hc the which for loss of value smallest the represents E inf E  ( VaR ( ( x CVaR 12 α ) = x  x { ( %ile z + ) / x  CVaR ( ) x : x , ) CVaR F CVaR [ ≥ DCVaR α 31 x  ( ( ( x VaR x z - ]  ) ( n sfruae as: formulated is and , − α E x > ) ( ( x x α E )) ) α α ( ( and x ( x } x )) )) loss ) [ Q DCVaR VaR R x ] [ of Q α x ( ] α [ ie ofiec level confidence a given , x , ( Q { x ) Y ) ] and ; } stedcso matrix. decision the is x , Ω = ,cnb translated be can ), CVaR loss α α ( x whereas , VaR ) a be can α (18) (20) (21) (19) 179 ( x ) . 180

Ω Ω into the probability function of loss (CG); then, the quantities CVaRα(CG ) and DCVaRα(CG ) R represent the level of risk and uncertainty associated to the solution [Q ] with an expected global Ω cost ECG = E(CG ), respectively.

3 Renewable generation selection, sizing and allocation

The practical aim of the MOO is to ﬁnd the optimal integration of DG in terms of selection, sizing and allocation of the different renewable generation units (PV and W). The corresponding decision R M R matrix [Q ] is contained in the matrix Q = [Q Q ] that stores the number and location of each type of power generator in the network. The MOO problem| consists in the simultaneous minimization of two objective functions, given by the indicators ECG and DCVaRα(CG).

3.1 Multi–objective optimization problem formulation

The general multi–objective optimization problem for all set of randomly generated operational scenarios Ω is formulated as follows:

min E CGΩ (22) R ( ) [Q ]i,j min DCVaR CGΩ (23) R α( ) [Q ]i,j s.t.

[Q]i,j Z∗ (2) ∈ [Q]i,j PAV j PF (24) Lmax i N Gj RG i ≤ X∈ X∈ MCS OPF([Q], Y , Ω) (13 -16) − { }

The meaning of each constraint is the following: (2) the decision variables [Q]i,j are non– negative integer numbers; (15) the ratio of total amount of average renewable power integrated in the network must be less or equal to the penetration factor PF; (13)–(16) all the power ﬂows equations must be satisﬁed.

3.1.1 Multi–objective differential evolution

The non–convex mixed–integer non–linear MOO problem under uncertainties is solved by the multi– objective differential evolution (MOO–DE) algorithm, integrating a fast non–dominated sorting procedure and crowded–comparison operator [53] into the original single objective DE [39], and

13 181 evaluating the objective functions by the developed MCS–OPF. The extension to MOO entails the integration of Pareto optimality concepts. In general terms, solving a MOO problem of the form:

min f1(X ), f2(X ),..., fm(X ) X { } s.t. X Λ ∈ n with at least two conﬂicting objectives functions (fi : ) implies to ﬁnd, within a set of acceptable solutions that belong to the non–empty feasibleℜ → region ℜ Λ n, the decision vectors ⊆ ℜ X Λ that satisfy the following [54]: ∈

X Λ/fi(X ) fi(X 0), i 1, 2, . . . , m and fi(X ) < fi(X 0) for at least one i ¬ ∈ ≤ ∀ ∈ { } ⇓ fi(X ) fi(X 0) i.e. X dominates X 0 ≺ The vector X is called a Pareto optimal solution and the Pareto front is deﬁned as the set f (X ) n such that X is Pareto optimal solution. The general MOO–DE algorithm is summarized{ as∈ follows:ℜ }

Initialization

Set the values of parameters: ◦ • NP: population size

• Genmax : maximum number of generations • COc: crossover coefﬁcient [0, 1] ∈ • F: differential variation ampliﬁcation factor [0, 2] ∈ Form the initial population POP0, randomly generating NP decision matrices (individuals) ◦ 0 0 0 0 X within the feasible space, POP = X1 ,..., Xk ,..., XNP {0 0 } 0 Evaluate the objective functions f1(Xk ),..., fNF (Xk ) for each individual Xk , where NF is ◦ the number of objective functions{ } Rank the individuals in POP0, applying a fast non–dominated sorting procedure with ◦ respect to the values of the objective functions 0 Compute and assign the crowding–distance value (dC ) to each individual in POP and ◦ sort, in ascending order, with respect to dC , the individuals belonging to the same non– domination–ranked group

Evolution loop

Set generations count index g = 1 ◦ g 0 Set POP = POP ◦ 14 182

While g Genmax (stopping criterion) ◦ ≤ Set a repository population RPOP as empty ◦ Trial loop g g For each individual Xk in POP , k 1, . . . , NP ∀ ∈ { } • Sample from the uniform distribution three integer indexes in 1,..., NP such that { g g }g k1 k2 k3 k and choose the corresponding three individuals X , X , X = = = k1 k2 k3 6 6 6 g • Generate a mutant individual XMk according to the following mutation operator:

g g g g XMk = Xk + F(Xk Xk ) (25) 1 2 − 3 g • Apply a crossover operator, initializing a randomly generated vector XCk , whose di- g g mensionality n is the same as that of Xk and each coordinate xck,i follows a uniform distribution with outcome in [0, 1] i 1,..., n . In addition, generate randomly an ∀ ∈ { } integer index i∗ in 1,..., n from a uniform distribution to ensure that at least one { g } g coordinate from XMk is exchanged to form trial individual XTk , whose coordinates g x tk,i are deﬁned as follows:

g g g xmk,i if xck,i Cco or i = i∗ x t = ≤ (26) k,i  g g xk,i if xck,i > Cco and i = i∗  6 • Evaluate the objective functions for the trial individual f XT g ,..., f XT g ; if XT g  1( k ) NF ( k ) k g g g g { g g } g dominates Xk , i.e., f (XTk ) f (Xk ) , XTk replaces Xk in POP , otherwise retain Xk g { g ≺ } in POP and save XTk in the repository population RPOP Set a combined population UPOP as POP g RPOP and rank the individuals in UPOP, ◦ applying a fast non–dominated sorting procedure∪

Compute and assign the crowding–distance value dC to each individual in UPOP and ◦ sort, in descending order with respect to dC , the individuals belonging to the same non– domination–ranked group g g Set POP as the ﬁrst NP individuals of the ranked and sorted population UPOP, POP = ◦ Xk : Xk UPOP, k 1, . . . , NP { ∈ ∈ { }} g If the stopping criterion is reached return POP , otherwise set g = g + 1. ◦ In correspondence to the nomenclature used in the proposed framework, the process of searching the set of non–dominated solutions carried out by the MOO–DE MCS–OPF is presented schematically in Figure 2.

15 183

Set the values of NP, Genmax , COc , F

R Generate randomly NP individuals [Q ], according to constraints (2) and (24), to 0 R 0 R 0 R 0 form the initial population POP = [Q ]1,..., [Q ]k,..., [Q ]NP { }

Ω Ω Evaluate the objective functions E(CG ) (22) and DCVaRα(CG ) (23) for each R 0 0 0 M R 0 individual [Q ]k through MCS-OPF([Q]k, [Y ], Ω) with [Q]k = [Q [Q ]k] |

g 0 Set generations count index g = 1 and POP = POP

Set k = 1 and repository population RPOP as empty

R g g From the individual [Q ]k in POP , k 1, . . . , NP , generate a trial individual R g ∀ ∈ { } [QT ]k by applying mutation (25) and crossover (26) operators and evaluate the Ω Ω g objective functions E(CG ) and DCVaRα(CG ) through MCS-OPF([QT]k , [Y ], Ω) g M R g with [QT]k = [Q [QT ]k ] |

Set k = k + 1

no Is k > NP?

yes Set the combined population UPOP as POP g RPOP, apply fast non–dominated sorting procedure on UPOP, within each non–domination–ranked∪ group, compute and

assign the crowding–distance values dC and sort the individuals in descending order

g R R Set POP = [Q ]k/[Q ]k UPOP, k 1, . . . , NP { ∈ ∈ { }}

Set g = g + 1 no Is g > Genmax ?

yes Return POP g and select the best front as the optimal set of non–dominated solutions

Figure 2: Flow chart of the proposed MOO–DE framework

16 184

4 Case study

We consider the IEEE 30 bus sub-transmission and distribution test system, a portion of the Midwest- ern U.S. electric power system which presents relevant characteristics of interest for the analysis, e.g., the presence of bulk–power supply spots different in type and with comparatively low and high nodal load proﬁles. An important consideration is that we neglect the synchronous condensers, given the DC assumptions made in the proposed framework for the resolution of the OPF problem.

4.1 Network description

The network consists of n = 30 nodes, a mesh deployment of 41 T&D lines and 2 transformers or bulk–power supply spots, as shown in Figure 3. The reference apparent power is Sre f = 100 (MV A).

Figure 3: IEEE 30 bus sub–transmission and distribution test system diagram

Table 1 summarizes the characteristics and technical data of the T&D lines, speciﬁcally: the indexes of the pair of nodes that they connect i, i , the susceptance values B i,i , power rating ( 0) ( 0) Pmax , failure λF and repair λR rates and operating cost COv i,i . (i,i ) (i,i ) (i,i ) ( ) 0 0 0 0

17 185

Table 1: T&D lines characteristics and technical data [40, 55, 56]

i i B i,i (p.u.) Pmax (p.u.) λF (n h) λR (n h) COv i,i ($ MWh) 0 ( ) (i,i ) (i,i ) / (i,i ) / ( ) / 0 0 0 0 0 1 2 17.24 1.30 2.85e 04 6.67e 02 4.95 1 3 5.41 1.30 2.85e−04 6.67e−02 1.61 2 4 5.75 0.65 4.28e−04 1.00e−01 2.61 3 4 26.32 1.30 2.85e−04 6.67e−02 5.97 2 5 5.05 1.30 2.85e−04 6.67e−02 2.53 2 6 5.68 0.65 4.28e−04 1.00e−01 3.68 4 6 24.39 0.90 3.73e−04 8.72e−02 6.06 5 7 8.62 0.70 4.17e−04 9.74e−02 0.70 6 7 12.20 1.30 2.85e−04 6.67e−02 0.93 6 8 23.81 0.32 5.01e−04 1.17e−01 0.19 6 9 4.81 0.65 4.28e−04 1.00e−01 5.21 6 10 1.80 0.32 5.01e−04 1.17e−01 4.14 9 10 9.09 0.65 4.28e−04 1.00e−01 8.78 9 11 4.81 0.65 4.28e−04 1.00e−01 4.81 4 12 3.91 0.65 4.28e−04 1.00e−01 2.93 12 13 7.14 0.65 4.28e−04 1.00e−01 1.41 12 14 3.91 0.32 5.01e−04 1.17e−01 11.10 12 15 7.69 0.32 5.01e−04 1.17e−01 6.66 14 15 5.00 0.16 5.36e−04 1.25e−01 13.70 12 16 5.03 0.32 5.01e−04 1.17e−01 13.70 10 17 11.76 0.32 5.01e−04 1.17e−01 13.20 16 17 5.18 0.16 5.36e−04 1.25e−01 8.71 15 18 4.57 0.16 5.36e−04 1.25e−01 3.46 18 19 7.75 0.16 5.36e−04 1.25e−01 12.20 10 20 4.78 0.32 5.01e−04 1.17e−01 5.40 19 20 14.71 0.32 5.01e−04 1.17e−01 3.52 10 21 13.33 0.32 5.01e−04 1.17e−01 11.50 10 22 6.67 0.32 5.01e−04 1.17e−01 9.76 21 22 41.67 0.32 5.01e−04 1.17e−01 8.08 15 23 4.95 0.16 5.36e−04 1.25e−01 1.68 22 24 5.59 0.16 5.36e−04 1.25e−01 7.82 23 24 3.70 0.16 5.36e−04 1.25e−01 16.40 24 25 3.04 0.16 5.36e−04 1.25e−01 7.98 25 26 2.63 0.16 5.36e−04 1.25e−01 9.13 25 27 4.78 0.16 5.36e−04 1.25e−01 5.56 6 28 16.67 0.32 5.01e−04 1.17e−01 1.15 8 28 5.00 0.32 5.01e−04 1.17e−01 0.36 27 28 2.53 0.65 4.28e−04 1.00e−01 6.17 27 29 2.41 0.16 5.36e−04 1.25e−01 14.40 27 30 1.66 0.16 5.36e−04 1.25e−01 14.00 29 30 2.21 0.16 5.36e−04 1.25e−01 11.40 − −

18 186

The nodal power demands are built from the load data given in [40] and reported in Figure 4 as daily proﬁles (accumulated according to the node indexes i), normally distributed on each hour t with mean µi,t and standard deviation σi,t . The technical data and uncertain model parameters of the different types of bulk–power suppliers and DG technologies, available to be integrated into the network, are given in Table 2. For the present case study, we consider that the number of photovoltaic cells per PV generation unit is nc = 20000 and that the region covered by the system is such that the solar irradiation and wind speed conditions are uniform in the whole region, i.e., the values of the parameters of the corresponding Beta and Weibull distributions are taken equal for all nodes. The renewable power penetration factor is set to PF = 30% (constraint (24) in the MOO problem).

Figure 4: Accumulated mean (A) and standard deviation (B) values of nodal load daily proﬁles

Table 3 reports the failure and repair rates, λF and λR, respectively, the investment and operating costs CI j + CO f j and the variable operating cost of the different types of power generators.

Concerning the network economics, the maximum value of the energy price is EPmax = 100 ($/MWh) [41,42,47] and the corresponding highest value of total demand ΣLmaxi (MW) is set to 445 (MW). The load shedding cost CLS ($/MWh) is considered as the maximum energy price. The horizon of analysis or lifetime of the project is 30 years, in which the investment and operating costs are hourly prorated. The conﬁdence level or α-percentile considered to estimate the values

CVaRα is 75%, arbitrarily chosen.

19 187

Table 2: Power generators technical data and uncertain model parameters [20, 41, 46, 50, 57, 58]

Type Technical parameters Distribution parameters

Pmax1 (MW) Normal µ1 Normal σ1 G1 340 300 18.25 MG

Pmax2 (MW) Normal µ2 Normal σ2 G2 50 42.5 5

Table 3: Power generators failure and repair rates and costs [20, 41, 46, 57–59]

Type λF (n/h) λR (n/h) CI + CO f (M$/u) COv ($/MWh)

G1 5.13e 04 2.77e 02 29.32 MG − − − G2 6.84e 04 4.16e 02 8.92 − − − PV 6.27e 04 1.30e 02 2.20 9.69 RG − − W 3.42e 04 9.00e 03 1.85 11.05 − −

Finally, Table 4 summarizes the main parameters set for the general MOO DE and MCS–OPF framework.

Table 4: MOO–DE and MCS–OPF parameters

Parameter Nomenclature Value Population size NP 100

Maximum n◦ of generations Genmax 600 Crossover coefﬁcient COc 0.1 Differential variation ampliﬁcation factor F 1

N◦ of MCS–OPF scenarios NS 24000 Scenario duration (h) t∆ 1

4.2 Results and discussion

The Pareto front resulting from the MOO–DE MCS–OPF is presented in Figure 5. The entire last generation population at convergence is shown by gray squares and the non–dominated solutions are the blue bullets. The base case (MG) in which no DG is integrated in the network is also shown R as dark gray triangle. Each solution in the Pareto Front corresponds to a decision matrix [Q ] that indicates the number of units and locations of the different types of DG technologies integrated in the network.

20 188

8.00 8.00

PF Dominated solutions 7.75 7.75 MG (no DG case) ) h / 7.50 $

k 7.50 (

) G C (

R 7.25 a 7.25 V C D

7.00 6.5000 7.00

6.3750

6.75 6.75 -12.50 -12.00 -11.506.2500 -11.00 -10.50 -12.50 -12.00 -11.50 -6.0000-11.00-5.9375 -10.50-5.8750 ECG (k$/h)

Figure 5: Set of non–dominated solutions: Pareto front

Recalling the relation given by equation (21), DCVaRα(x) = CVaRα(x E(x)), it is possible to draw a map of iso–CVaR curves in the plot of non–dominated solutions and so,− to include risk into the trade–off between expected performance and uncertainty, represented by ECG and DCVaRα(CG), respectively. Then, we can ﬁnd the compromised solution, namely QR min , that minimizes [ ]CVaR(CG) risk as reported in Figure 6(A). The reciprocal case, i.e., a map of iso–DCVaR curves is drawn in the distribution of non–dominated solutions plotted as ECG vs DCVaRα(CG), and it is shown in Figure 6(B). We look to three representative non–dominated solutions for the analysis: those with minimum values of ECG and DCVaR CG , denoted QR min and QR min respectively, and α( ) [ ]ECG [ ]DCVaR(CG) QR min which, as mentioned earlier, minimizes risk. [ ]CVaR(CG) R min R min R min In Figure 6, we can observe that the three solutions of interest, [Q ]ECG, [Q ]CVaR and [Q ]DCVaR, lead to considerable improvements in expected performance and risk with respect to the base case MG, in which no renewable generation is integrated into the network. However, the level of uncertainty in the ECG estimation is increased, in all DG–integrated solutions, because of their stochasticity. Even so, in comparison to the MG case, the increase in the level of uncertainty for all DG–integrated cases (on average 1.067 (k$/h)) is much less than the gain in both expected performance and risk (on average 6.035 and 4.967 (k$/h)), respectively). This fact can be seen also in the empirical CG probability− density functions− (pdf) shown in Figure 7.

21 1hus,bcuehge eeso oe ead ml oeeeg odad hrfr,more therefore, hours, and, 21 sold to energy more 18 imply and demands 13 power to of 10 levels intervals higher because the hours), in 21 place taking load of ranges high two and hours 6 epciey hs h etpa ftedsrbtoscrepnst h ihs ag flas(8to (18 loads of range highest the to corresponds distributions the and of 23 peak which between left 4) night, the (Figure the Thus, profiles respectively. during load range daily demand the power low of a characteristics the ranges: to important due three is present This peaks. main three rfiso eaievle of values negative or profits utemr,i a eifre rmFgr ht ngnrl all general, in that, 7 Figure from inferred be can it Furthermore,

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0.08 DCVaRa(CG) = 6.379 (k$/h) ECG = -5.937 (k$/h) CVaR (CG) = 0.442 (k$/h) a Case MG 0.06 1.2E-03

0.04 8.0E-04

frequency 4.0E-04 0.02 0.0E+00 0 10 20 30 40 50 60 0

0.048 -30 DCVaR-20 a(CG-10) = 7.8620 (k$/h) 10 20 30 40 50 60 ECG = -12.281 (k$/h) CVaRa(CG) = -4.419 (k$/h) R min Ca se [Q ] ECG 0.036 1.2E-03

0.024 8.0E-04

frequency 4.0E-04 0.012 0.0E+00 0 10 20 30 40 50 60 0 -30 -20 -10 0 10 20 30 40 50 60 CG (k$/h)

Figure 7: Empirical CG probability density functions

23 191 highest range of loads (10 to 13 hours) and the right peak to the low range of power demand. As mentioned before, the three DG–integrated network cases improve the cost profiles because the usage of power is transferred from the bulk–power suppliers (MG) to the renewable generators (RG) as summarized in Table 5, presenting the ratio of power usage defined as the proportion of power used to satisfy the loads, determined by the MCS–OPF, over the power available. According to the IEEE 30 bus test systems information, the bulk–power supply arriving at node 1, G1, comes from a conventional coal–fired power plant whereas G2 is supplied by a hydro–power plant. The nature of the source of power supply conditions the operating costs, in particular, G1 that presents the higher variable operating costs (Table 3). Nevertheless, the ratio of power usage associated to G2 is in all cases 100%, even though its operating cost is higher than the renewable. This is because no investment is being paid for G2, contrary to the DG technologies (PV and W). In this view, considering investment, PV and W are more convenient than the coal–fired power supply G1 but not more than the hydro–power supply G2.

Table 5: Ratio of power usage by type of generator

Case G1 G2 PVW MG 77.93% 100.00% − − ECGmin 41.88% 100.00% 99.96% 99.94% CVaRCGmin 40.38% 100.00% 99.95% 99.42% DCVaRCGmin 40.71% 100.00% 100.00% 98.73%

The extreme CG scenarios encountered in the tail of the distributions are mainly produced by the occurrence of failures in the components of the system, power generators and T&D lines. In Figure 7, we notice the stability of the CG to these non–desirable events. Since the cost objective function to be minimized in the OPF considers a load shedding cost, the occurrence of failures in the components, interrupting the power supply and/or the ability of distribute it, will impact the CG function depending on how much centralized is the power supply. Focusing on the MG base case, the power supply and its distribution depend on two generators and the T&D lines connected to them: the reliability of power supply is determined by few components and so, the eventual losses of functionality of these components can lead to high amounts of non–satisﬁed demands. This is precisely the effect observed in the tails of the CG distributions and, in particular, we see that distributing and diversifying generation helps to improve the risk impacts from multiple failures in the network, even if the number of generation units in the system is increased and with it, the overall absolute likelihood of occurrence of failures.

24 192

300 50 P (MW) Case [QR ]min AVij, ECG 40 PV 30 G1 W 20 G2 10

0 0 300 50 P (MW) Case [QR ]min AVij, CVaR() CG 40 PV 30 G1 W 20 G2 10

0 0 300 50 P (MW) Case [QR ]min AVij, DCVaR() CG 40 120 -120 PV 30 G1 W 80 20-80 G2 10 40 -40 0 0 0 0

-40 40

-80 80 L (MW) AVi -120 120 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 node i

Figure 8: Total average renewable power installed. Cases ECGmin (A), CVaRCGmin (B) and DCVaRCGmin (C)

Figure 8 shows the total average DG power integrated in the network for the three solutions under analysis and the corresponding obtained values of ECG and CVaRα. It can be pointed out that in all cases the DG power installed is almost equal to the limit value set by the penetration factor PF. In the present case, a maximum of 30% is accepted, of average DG power installed over the maximum aggregated load in the system. This leads to an approximated limit of 135.5 (MW). Furthermore, in general terms moving along the non–dominated solutions, starting from the R min one that minimizes ECG ([Q ]ECG), passing to the compromising one that minimizes CVaRα(CG) ( QR min ) and ending with ( QR min ) that minimizes DCVaR CG , the amount of PV [ ]CVaR(CG) [ ]DCVaR(CG) α( ) power integrated in the network decreases progressively, being replaced by W power. Then, we infer that the more PV power generators are integrated, the better expected global cost performance is achieved. This seems physically coherent, taking into account that the average power outputs delivered by one generation unit of PV and W technologies are 1.07 and 0.93 (MW), respectively,

25 193 and the cost performance beneﬁts comparatively more from the PV MW h sold during the daylight interval within the two peaks ranges of power demand. Nevertheless, the amount of average integrated PV power is invariably smaller than W power, this is because PV generation units do not supply during the night interval, making convenient to integrate always a certain amount of W power and, thus, to avoid resorting to the more expensive coal–ﬁred power supplier G1. Moreover, it is precisely this lack of PV generation during the night interval that strongly conditions the trade–off between expected performance and uncertainty, ECG vs DCVaRα(CG), i.e., the more PV power is integrated, the better expected global cost performance is achieved but the more uncertain is its estimation.

PV W

ECG CVaRa (CG) 150 -4.0

100 -5.0 ) G

C (MW) (

, ij a (k$/h)

R AV

a P V

50 C -11.5

D 

0 -12.5 A B C

Figure 9: Nodal average power by type of generator

Concerning the compromising solution QR min that minimizes risk, as it was derived from [ ]CVaR(CG) the empirical CG distributions, the risk associated to a solution depends mainly to the occurrence of extreme non–desired events. Improvements in risk performance can, then, be achieved by solutions for which the allocation of DG power generation units decentralizes and diversifies to a large extent the supply. This insight is noticeable in Figure 9 that reports the nodal average power by type of R min R min generation for the three solutions of interest. For both extreme solutions, [Q ]ECG and [Q ]DCVaR, the tendency is to integrate localized sources of renewable DG at two identifiable portions of the network, in the region close to nodes 2, 5, 7 and 8 of the sub–transmission portion of the network, and nodes 17, 19 and 21 in the distribution part, favoring the sub–transmission portion which presents higher and non homogeneous nodal load profiles. In a different manner, the solution QR min [ ]CVaR(CG) presents a more homogeneous deployment of DG power, allocating comparable generation capacities in both sub–transmission and distribution parts of the network.

26 194

5 Conclusions

We have presented a multi–objective optimization framework for the integration of renewable distributed generation into an electric power network. Multiple uncertain operational inputs are taken into consideration: the inherent uncertain behavior of renewable energy sources and power demands, as well as the occurrence of failures of components. For managing the uncertainty and risk associated to the achievement of a certain level of expected global cost performance, we have introduced the conditional–value–at–risk deviation measure, which allows trading off the level of uncertainty and, given the axiomatic relation to the conditional–value–at–risk, enables conjointly the trade–off of risk by constructing an iso–risk map in the non–dominated set of solutions. The proposed framework integrates the multi–objective differential evolution as a search engine, Monte Carlo simulation to randomly generate realizations of the uncertain operational scenarios and optimal power ﬂow to evaluate the network response. The optimization is done to simultaneously minimize the expected value of the global costs and the respective conditional–value–at–risk deviation.

A case study has been analyzed, based on the IEEE 30 bus sub–transmission and distribution test system. The results obtained show the capability of the framework to identify Pareto optimal solutions of renewable DG units allocations. Integrating the conditional value–at–risk deviation into the framework has shown effectively the possibility of optimizing expected performances while controlling the uncertainty and risk, analyzing, in addition, the contribution of each type of renewable DG technology on the level of uncertainty associated to the outcome performance of the optimal solutions and the importance of the deployment of the renewable generation capacity to lower the risk of incurring in non–desirable extreme scenarios. In this view, a complete and comprehensible spectrum of information can be supplied in support of speciﬁc preferences of the decision makers for their decision tasks.

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